uncertainty theory

Uncertainty Theory

Third Edition

Baoding LiuUncertainty Theory Laboratory

Department of Mathematical Sciences

Tsinghua University

Beijing 100084, China

[email protected]

http://orsc.edu.cn/liu

3rd Edition c© 2008 by UTLAB

Japanese Translation Version c© 2008 by WAP

2nd Edition c© 2007 by Springer-Verlag Berlin

1st Edition c© 2004 by Springer-Verlag Berlin

Reference to this book should be made as follows:Liu B, Uncertainty Theory, 3rd ed., http://orsc.edu.cn/liu/ut.pdf


http://orsc.edu.cn/utlab

mailto:[email protected]


http://orsc.edu.cn/utlab

http://orsc.edu.cn/liu/ut.pdf

Contents

Preface ix

1 Probability Theory 11.1 Probability Space . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Probability Distribution . . . . . . . . . . . . . . . . . . . . . 71.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 131.6 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.8 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.9 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 261.10 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.11 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.12 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.13 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 351.14 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 401.15 Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . 431.16 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . 471.17 Stochastic Differential Equation . . . . . . . . . . . . . . . . 50

2 Credibility Theory 532.1 Credibility Space . . . . . . . . . . . . . . . . . . . . . . . . . 532.2 Fuzzy Variables . . . . . . . . . . . . . . . . . . . . . . . . . 632.3 Membership Function . . . . . . . . . . . . . . . . . . . . . . 652.4 Credibility Distribution . . . . . . . . . . . . . . . . . . . . . 692.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.6 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 792.7 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 802.8 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.9 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.10 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 962.11 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

vi Contents

2.12 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.13 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.14 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 1102.15 Conditional Credibility . . . . . . . . . . . . . . . . . . . . . 1142.16 Fuzzy Process . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.17 Fuzzy Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.18 Fuzzy Differential Equation . . . . . . . . . . . . . . . . . . . 128

3 Chance Theory 1293.1 Chance Space . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.2 Hybrid Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1373.3 Chance Distribution . . . . . . . . . . . . . . . . . . . . . . . 1433.4 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 1463.5 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.6 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.7 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.8 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 1533.9 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.10 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.11 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.12 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.13 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 1603.14 Conditional Chance . . . . . . . . . . . . . . . . . . . . . . . 1643.15 Hybrid Process . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.16 Hybrid Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 1713.17 Hybrid Differential Equation . . . . . . . . . . . . . . . . . . 174

4 Uncertainty Theory 1774.1 Uncertainty Space . . . . . . . . . . . . . . . . . . . . . . . . 1774.2 Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . . 1814.3 Identification Function . . . . . . . . . . . . . . . . . . . . . 1844.4 Uncertainty Distribution . . . . . . . . . . . . . . . . . . . . 1854.5 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . 1874.6 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.7 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.8 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934.9 Identical Distribution . . . . . . . . . . . . . . . . . . . . . . 1934.10 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . 1944.11 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.12 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.13 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.14 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . 2004.15 Conditional Uncertainty . . . . . . . . . . . . . . . . . . . . . 2014.16 Uncertain Process . . . . . . . . . . . . . . . . . . . . . . . . 2054.17 Uncertain Calculus . . . . . . . . . . . . . . . . . . . . . . . 209

Contents vii

4.18 Uncertain Differential Equation . . . . . . . . . . . . . . . . 211

A Measurable Sets 213

B Classical Measures 216

C Measurable Functions 218

D Lebesgue Integral 221

E Euler-Lagrange Equation 224

F Maximum Uncertainty Principle 225

G Uncertainty Relations 226

Bibliography 229

List of Frequently Used Symbols 244

Index 245

viii Contents

Preface

There are various types of uncertainty in the real world. Randomness is abasic type of objective uncertainty, and probability theory is a branch ofmathematics for studying the behavior of random phenomena. The studyof probability theory was started by Pascal and Fermat (1654), and an ax-iomatic foundation of probability theory was given by Kolmogoroff (1933) inhis Foundations of Probability Theory. Probability theory has been widelyapplied in science and engineering. Chapter 1 will provide the probabilitytheory.

Fuzziness is a basic type of subjective uncertainty initiated by Zadeh(1965). Credibility theory is a branch of mathematics for studying the be-havior of fuzzy phenomena. The study of credibility theory was started byLiu and Liu (2002), and an axiomatic foundation of credibility theory wasgiven by Liu (2004) in his Uncertainty Theory. Chapter 2 will introduce thecredibility theory.

Sometimes, fuzziness and randomness simultaneously appear in a sys-tem. A hybrid variable was proposed by Liu (2006) as a tool to describe thequantities with fuzziness and randomness. Both fuzzy random variable andrandom fuzzy variable are instances of hybrid variable. In addition, Li andLiu (2007) introduced the concept of chance measure for hybrid events. Afterthat, chance theory was developed steadily. Essentially, chance theory is ahybrid of probability theory and credibility theory. Chapter 3 will offer thechance theory.

In order to deal with general uncertainty, Liu (2007) founded an uncer-tainty theory in his Uncertainty Theory, and had it become a branch ofmathematics based on normality, monotonicity, self-duality, and countablesubadditivity axioms. Probability theory, credibility theory and chance the-ory are three special cases of uncertainty theory. Chapter 4 is devoted to theuncertainty theory.

For this new edition the entire text has been totally rewritten. Moreimportantly, uncertain process and uncertain calculus as well as uncertaindifferential equations have been added.

The book is suitable for mathematicians, researchers, engineers, design-ers, and students in the field of mathematics, information science, operationsresearch, industrial engineering, computer science, artificial intelligence, andmanagement science. The readers will learn the axiomatic approach of un-certainty theory, and find this work a stimulating and useful reference.

x Preface

Baoding LiuTsinghua Universityhttp://orsc.edu.cn/liu

March 5, 2008


Chapter 1

Probability Theory

Probability measure is essentially a set function (i.e., a function whose ar-gument is a set) satisfying normality, nonnegativity and countable additivityaxioms. Probability theory is a branch of mathematics for studying the be-havior of random phenomena. The emphasis in this chapter is mainly onprobability space, random variable, probability distribution, independence,identical distribution, expected value, variance, moments, critical values, en-tropy, distance, convergence almost surely, convergence in probability, conver-gence in mean, convergence in distribution, conditional probability, stochasticprocess, renewal process, Brownian motion, stochastic calculus, and stochas-tic differential equation. The main results in this chapter are well-known.For this reason the credit references are not provided.

1.1 Probability Space

Let Ω be a nonempty set, and A a σ-algebra over Ω. If Ω is countable, usuallyA is the power set of Ω. If Ω is uncountable, for example Ω = [0, 1], usuallyA is the Borel algebra of Ω. Each element in A is called an event.

In order to present an axiomatic definition of probability, it is necessaryto assign to each event A a number PrA which indicates the probabilitythat A will occur. In order to ensure that the number PrA has certainmathematical properties which we intuitively expect a probability to have,the following three axioms must be satisfied:

Axiom 1. (Normality) PrΩ = 1.

Axiom 2. (Nonnegativity) PrA ≥ 0 for any event A.

Axiom 3. (Countable Additivity) For every countable sequence of mutually

2 Chapter 1 - Probability Theory

disjoint events Ai, we have

Pr

∞⋃i=1

Ai

=

∞∑i=1

PrAi. (1.1)

Definition 1.1 The set function Pr is called a probability measure if it sat-isfies the normality, nonnegativity, and countable additivity axioms.

Example 1.1: Let Ω = ω1, ω2, · · · , and let A be the power set of Ω.Assume that p1, p2, · · · are nonnegative numbers such that p1 + p2 + · · · = 1.Define a set function on A as

PrA =∑

ωi∈A

pi, A ∈ A. (1.2)

Then Pr is a probability measure.

Example 1.2: Let Ω = [0, 1] and let A be the Borel algebra over Ω. If Pr isthe Lebesgue measure, then Pr is a probability measure.

Theorem 1.1 Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr aprobability measure. Then we have(a) Pr∅ = 0;(b) Pr is self-dual, i.e., PrA+ PrAc = 1 for any A ∈ A;(c) Pr is increasing, i.e., PrA ≤ PrB whenever A ⊂ B.

Proof: (a) Since ∅ and Ω are disjoint events and ∅ ∪ Ω = Ω, we havePr∅+ PrΩ = PrΩ which makes Pr∅ = 0.

(b) Since A and Ac are disjoint events and A∪Ac = Ω, we have PrA+PrAc = PrΩ = 1.

(c) Since A ⊂ B, we have B = A ∪ (B ∩ Ac), where A and B ∩ Ac aredisjoint events. Therefore PrB = PrA+ PrB ∩Ac ≥ PrA.

Probability Continuity Theorem

Theorem 1.2 (Probability Continuity Theorem) Let Ω be a nonempty set,A a σ-algebra over Ω, and Pr a probability measure. If A1, A2, · · · ∈ A andlimi→∞Ai exists, then

limi→∞

PrAi = Pr

limi→∞

Ai

. (1.3)

Proof: Step 1: Suppose Ai is an increasing sequence. Write Ai → A andA0 = ∅. Then Ai\Ai−1 is a sequence of disjoint events and

∞⋃i=1

(Ai\Ai−1) = A,k⋃

i=1

(Ai\Ai−1) = Ak

Section 1.1 - Probability Space 3

for k = 1, 2, · · · Thus we have

PrA = Pr ∞⋃

i=1

(Ai\Ai−1)

=∞∑

i=1

Pr Ai\Ai−1

= limk→∞

k∑i=1

Pr Ai\Ai−1 = limk→∞

Pr

k⋃i=1

(Ai\Ai−1)

= limk→∞

PrAk.

Step 2: If Ai is a decreasing sequence, then the sequence A1\Ai isclearly increasing. It follows that

PrA1 − PrA = Pr

limi→∞

(A1\Ai)

= limi→∞

Pr A1\Ai

= PrA1 − limi→∞

PrAi

which implies that PrAi → PrA.Step 3: If Ai is a sequence of events such that Ai → A, then for each

k, we have∞⋂

i=k

Ai ⊂ Ak ⊂∞⋃

i=k

Ai.

Since Pr is increasing, we have

Pr

∞⋂i=k

Ai

≤ PrAk ≤ Pr

∞⋃i=k

Ai

.

Note that∞⋂

i=k

Ai ↑ A,∞⋃

i=k

Ai ↓ A.

It follows from Steps 1 and 2 that PrAi → PrA.

Probability Space

Definition 1.2 Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr aprobability measure. Then the triplet (Ω,A,Pr) is called a probability space.

Example 1.3: Let Ω = ω1, ω2, · · · , A the power set of Ω, and Pr aprobability measure defined by (1.2). Then (Ω,A,Pr) is a probability space.

Example 1.4: Let Ω = [0, 1], A the Borel algebra over Ω, and Pr theLebesgue measure. Then ([0, 1],A,Pr) is a probability space, and sometimesis called Lebesgue unit interval. For many purposes it is sufficient to use itas the basic probability space.


Product Probability Space

Let (Ωi,Ai,Pri), i = 1, 2, · · · , n be probability spaces, and Ω = Ω1 × Ω2 ×· · · × Ωn, A = A1 × A2 × · · · × An. Note that the probability measuresPri, i = 1, 2, · · · , n are finite. It follows from the product measure theoremthat there is a unique measure Pr on A such that

PrA1 ×A2 × · · · ×An = Pr1A1 × Pr2A2 × · · · × PrnAn

for any Ai ∈ Ai, i = 1, 2, · · · , n. This conclusion is called the product proba-bility theorem. The measure Pr is also a probability measure since

PrΩ = Pr1Ω1 × Pr2Ω2 × · · · × PrnΩn = 1.

Such a probability measure is called the product probability measure, denotedby Pr = Pr1 × Pr2 × · · · × Prn.

Definition 1.3 Let (Ωi,Ai,Pri), i = 1, 2, · · · , n be probability spaces, andΩ = Ω1×Ω2×· · ·×Ωn, A = A1×A2×· · ·×An, Pr = Pr1×Pr2× · · ·×Prn.Then the triplet (Ω,A,Pr) is called the product probability space.

Infinite Product Probability Space

Let (Ωi,Ai,Pri), i = 1, 2, · · · be an arbitrary sequence of probability spaces,and

Ω = Ω1 × Ω2 × · · · , A = A1 ×A2 × · · · (1.4)

It follows from the infinite product measure theorem that there is a uniqueprobability measure Pr on A such that

Pr A1 × · · · ×An × Ωn+1 × Ωn+2 × · · · = Pr1A1 × · · · × PrnAn

for any measurable rectangle A1 × · · · × An × Ωn+1 × Ωn+2 × · · · and alln = 1, 2, · · · The probability measure Pr is called the infinite product ofPri, i = 1, 2, · · · and is denoted by

Pr = Pr1 × Pr2 × · · · (1.5)

Definition 1.4 Let (Ωi,Ai,Pri), i = 1, 2, · · · be probability spaces, and Ω =Ω1 × Ω2 × · · · , A = A1 × A2 × · · · , Pr = Pr1×Pr2× · · · Then the triplet(Ω,A,Pr) is called the infinite product probability space.

1.2 Random Variables

Definition 1.5 A random variable is a measurable function from a proba-bility space (Ω,A,Pr) to the set of real numbers, i.e., for any Borel set B ofreal numbers, the set

ξ ∈ B = ω ∈ Ω∣∣ ξ(ω) ∈ B (1.6)

is an event.

Section 1.2 - Random Variables 5

Example 1.5: Take (Ω,A,Pr) to be ω1, ω2 with Prω1 = Prω2 = 0.5.Then the function

ξ(ω) =

0, if ω = ω1

1, if ω = ω2

is a random variable.

Example 1.6: Take (Ω,A,Pr) to be the interval [0, 1] with Borel algebraand Lebesgue measure. We define ξ as an identity function from Ω to [0,1].Since ξ is a measurable function, it is a random variable.

Example 1.7: A deterministic number c may be regarded as a special ran-dom variable. In fact, it is the constant function ξ(ω) ≡ c on the probabilityspace (Ω,A,Pr).

Definition 1.6 A random variable ξ is said to be(a) nonnegative if Prξ < 0 = 0;(b) positive if Prξ ≤ 0 = 0;(c) continuous if Prξ = x = 0 for each x ∈ <;(d) simple if there exists a finite sequence x1, x2, · · · , xm such that

Pr ξ 6= x1, ξ 6= x2, · · · , ξ 6= xm = 0; (1.7)

(e) discrete if there exists a countable sequence x1, x2, · · · such that

Pr ξ 6= x1, ξ 6= x2, · · · = 0. (1.8)

Definition 1.7 Let ξ1 and ξ2 be random variables defined on the probabilityspace (Ω,A,Pr). We say ξ1 = ξ2 if ξ1(ω) = ξ2(ω) for almost all ω ∈ Ω.

Random Vector

Definition 1.8 An n-dimensional random vector is a measurable functionfrom a probability space (Ω,A,Pr) to the set of n-dimensional real vectors,i.e., for any Borel set B of <n, the set

ξ ∈ B =ω ∈ Ω

∣∣ ξ(ω) ∈ B

(1.9)

is an event.

Theorem 1.3 The vector (ξ1, ξ2, · · · , ξn) is a random vector if and only ifξ1, ξ2, · · · , ξn are random variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is a random vector on theprobability space (Ω,A,Pr). For any Borel set B of <, the set B × <n−1 isalso a Borel set of <n. Thus we have

ω ∈ Ω∣∣ ξ1(ω) ∈ B

=ω ∈ Ω

∣∣ ξ1(ω) ∈ B, ξ2(ω) ∈ <, · · · , ξn(ω) ∈ <

=ω ∈ Ω

∣∣ ξ(ω) ∈ B ×<n−1∈ A


which implies that ξ1 is a random variable. A similar process may prove thatξ2, ξ3, · · · , ξn are random variables.

Conversely, suppose that all ξ1, ξ2, · · · , ξn are random variables on theprobability space (Ω,A,Pr). We define

B =B ⊂ <n

∣∣ ω ∈ Ω|ξ(ω) ∈ B ∈ A.

The vector ξ = (ξ1, ξ2, · · · , ξn) is proved to be a random vector if we canprove that B contains all Borel sets of <n. First, the class B contains allopen intervals of <n because

ω ∈ Ω∣∣ ξ(ω) ∈

n∏i=1

(ai, bi)

=

n⋂i=1

ω ∈ Ω

∣∣ ξi(ω) ∈ (ai, bi)∈ A.

Next, the class B is a σ-algebra of <n because (i) we have <n ∈ B sinceω ∈ Ω|ξ(ω) ∈ <n = Ω ∈ A; (ii) if B ∈ B, then ω ∈ Ω|ξ(ω) ∈ B ∈ A, and

ω ∈ Ω∣∣ ξ(ω) ∈ Bc = ω ∈ Ω

∣∣ ξ(ω) ∈ Bc ∈ A

which implies that Bc ∈ B; (iii) if Bi ∈ B for i = 1, 2, · · · , then ω ∈ Ω|ξ(ω) ∈Bi ∈ A and

ω ∈ Ω∣∣ ξ(ω) ∈

∞⋃i=1

Bi

=

∞⋃i=1

ω ∈ Ω∣∣ ξ(ω) ∈ Bi ∈ A

which implies that ∪iBi ∈ B. Since the smallest σ-algebra containing allopen intervals of <n is just the Borel algebra of <n, the class B contains allBorel sets of <n. The theorem is proved.

Random Arithmetic

In this subsections, we will suppose that all random variables are defined on acommon probability space. Otherwise, we may embed them into the productprobability space.

Definition 1.9 Let f : <n → < be a measurable function, and ξ1, ξ2, · · · , ξnrandom variables defined on the probability space (Ω,A,Pr). Then ξ =f(ξ1, ξ2, · · · , ξn) is a random variable defined by

ξ(ω) = f(ξ1(ω), ξ2(ω), · · · , ξn(ω)), ∀ω ∈ Ω. (1.10)

Example 1.8: Let ξ1 and ξ2 be random variables on the probability space(Ω,A,Pr). Then their sum is

(ξ1 + ξ2)(ω) = ξ1(ω) + ξ2(ω), ∀ω ∈ Ω

Section 1.3 - Probability Distribution 7

and their product is

(ξ1 × ξ2)(ω) = ξ1(ω)× ξ2(ω), ∀ω ∈ Ω.

The reader may wonder whether ξ(ω1, ω2, · · · , ωn) defined by (1.9) is arandom variable. The following theorem answers this question.

Theorem 1.4 Let ξ be an n-dimensional random vector, and f : <n → < ameasurable function. Then f(ξ) is a random variable.

Proof: Assume that ξ is a random vector on the probability space (Ω,A,Pr).For any Borel set B of <, since f is a measurable function, f−1(B) is also aBorel set of <n. Thus we have

ω ∈ Ω∣∣ f(ξ(ω)) ∈ B

=ω ∈ Ω

∣∣ ξ(ω) ∈ f−1(B)∈ A

which implies that f(ξ) is a random variable.

1.3 Probability Distribution

Definition 1.10 The probability distribution Φ: < → [0, 1] of a randomvariable ξ is defined by

Φ(x) = Prω ∈ Ω

∣∣ ξ(ω) ≤ x. (1.11)

That is, Φ(x) is the probability that the random variable ξ takes a value lessthan or equal to x.

Example 1.9: Assume that the random variables ξ and η have the sameprobability distribution. One question is whether ξ = η or not. Gener-ally speaking, it is not true. Take (Ω,A,Pr) to be ω1, ω2 with Prω1 =Prω2 = 0.5. We now define two random variables as follows,

ξ(ω) =−1, if ω = ω1

1, if ω = ω2,η(ω) =

1, if ω = ω1

−1, if ω = ω2.

Then ξ and η have the same probability distribution,

Φ(x) =

0, if x < −1

0.5, if − 1 ≤ x < 11, if x ≥ 1.

However, it is clear that ξ 6= η in the sense of Definition 1.7.

Theorem 1.5 (Sufficient and Necessary Condition for Probability Distribu-tion) A function Φ : < → [0, 1] is a probability distribution if and only if it isan increasing and right-continuous function with

limx→−∞

Φ(x) = 0; limx→+∞

Φ(x) = 1. (1.12)


Proof: For any x, y ∈ < with x < y, we have

Φ(y)− Φ(x) = Prx < ξ ≤ y ≥ 0.

Thus the probability distribution Φ is increasing. Next, let εi be a sequenceof positive numbers such that εi → 0 as i → ∞. Then, for every i ≥ 1, wehave

Φ(x+ εi)− Φ(x) = Prx < ξ ≤ x+ εi.

It follows from the probability continuity theorem that

limi→∞

Φ(x+ εi)− Φ(x) = Pr∅ = 0.

Hence Φ is a right-continuous function. Finally,

limx→−∞

Φ(x) = limx→−∞

Prξ ≤ x = Pr∅ = 0,

limx→+∞

Φ(x) = limx→+∞

Prξ ≤ x = PrΩ = 1.

Conversely, it is known there is a unique probability measure Pr on the Borelalgebra over < such that Pr(−∞, x] = Φ(x) for all x ∈ <. Furthermore,it is easy to verify that the random variable defined by ξ(x) = x from theprobability space (<,A,Pr) to < has the probability distribution Φ.

Remark 1.1: Theorem 1.5 states that the identity function is a universalfunction for any probability distribution by defining an appropriate proba-bility space. In fact, there is a universal probability space for any probabilitydistribution by defining an appropriate function.

Theorem 1.6 Let Φ be a probability distribution. Then there is a randomvariable on Lebesgue unit interval ([0, 1],A,Pr) whose probability distributionis just Φ.

Proof: Define ξ(ω) = supx|Φ(x) ≤ ω on the Lebesgue unit interval. Thenξ is a random variable whose probability distribution is just Φ because

Prξ ≤ y = Prω∣∣ sup x|Φ(x) ≤ ω ≤ y

= Pr

ω∣∣ ω ≤ Φ(y)

= Φ(y)

for any y ∈ <.

Theorem 1.7 A random variable ξ with probability distribution Φ is(a) nonnegative if and only if Φ(x) = 0 for all x < 0;(b) positive if and only if Φ(x) = 0 for all x ≤ 0;(c) simple if and only if Φ is a simple function;(d) discrete if and only if Φ is a step function;(e) continuous if and only if Φ is a continuous function.

Section 1.3 - Probability Distribution 9

Proof: The parts (a), (b), (c) and (d) follow immediately from the definition.Next we prove the part (e). If ξ is a continuous random variable, thenPrξ = x = 0. It follows from the probability continuity theorem that

limy↑x

(Φ(x)− Φ(y)) = limy↑x

Pry < ξ ≤ x = Prξ = x = 0

which proves the left-continuity of Φ. Since a probability distribution isalways right-continuous, Φ is continuous. Conversely, if Φ is continuous,then we immediately have Prξ = x = 0 for each x ∈ <.

Definition 1.11 A continuous random variable is said to be (a) singular ifits probability distribution is a singular function; (b) absolutely continuous ifits probability distribution is an absolutely continuous function.

Probability Density Function

Definition 1.12 The probability density function φ: < → [0,+∞) of a ran-dom variable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy (1.13)

holds for all x ∈ <, where Φ is the probability distribution of the randomvariable ξ.

Let φ : < → [0,+∞) be a measurable function such that∫ +∞−∞ φ(x)dx = 1.

Then φ is the probability density function of some random variable becauseΦ(x) =

∫ x

−∞ φ(y)dy is an increasing and continuous function satisfying (1.12).

Theorem 1.8 Let ξ be a random variable whose probability density functionφ exists. Then for any Borel set B of <, we have

Prξ ∈ B =∫

B

φ(y)dy. (1.14)

Proof: Let C be the class of all subsets C of < for which the relation

Prξ ∈ C =∫

C

φ(y)dy (1.15)

holds. We will show that C contains all Borel sets of <. It follows from theprobability continuity theorem and relation (1.15) that C is a monotone class.It is also clear that C contains all intervals of the form (−∞, a], (a, b], (b,∞)and ∅ since

Prξ ∈ (−∞, a] = Φ(a) =∫ a

−∞φ(y)dy,

Prξ ∈ (b,+∞) = Φ(+∞)− Φ(b) =∫ +∞

b

φ(y)dy,


Prξ ∈ (a, b] = Φ(b)− Φ(a) =∫ b

a

φ(y)dy,

Prξ ∈ ∅ = 0 =∫∅φ(y)dy

where Φ is the probability distribution of ξ. Let F be the algebra consisting ofall finite unions of disjoint sets of the form (−∞, a], (a, b], (b,∞) and ∅. Notethat for any disjoint sets C1, C2, · · · , Cm of F and C = C1 ∪ C2 ∪ · · · ∪ Cm,we have

Prξ ∈ C =m∑

j=1

Prξ ∈ Cj =m∑

j=1

∫Cj

φ(y)dy =∫

C

φ(y)dy.

That is, C ∈ C. Hence we have F ⊂ C. Since the smallest σ-algebra containingF is just the Borel algebra of <, the monotone class theorem implies that Ccontains all Borel sets of <.

Some Special Distributions

Uniform Distribution: A random variable ξ has a uniform distribution ifits probability density function is defined by

φ(x) =

1

b− a, if a ≤ x ≤ b

0, otherwise(1.16)

denoted by U(a, b), where a and b are given real numbers with a < b.

Exponential Distribution: A random variable ξ has an exponential dis-tribution if its probability density function is defined by

φ(x) =

1β

exp(−xβ

), if x ≥ 0

0, if x < 0(1.17)

denoted by EXP(β), where β is a positive number.

Normal Distribution: A random variable ξ has a normal distribution ifits probability density function is defined by

φ(x) =1

σ√

2πexp

(− (x− µ)2

2σ2

), x ∈ < (1.18)

denoted by N (µ, σ2), where µ and σ are real numbers.

Section 1.4 - Independence 11

Joint Probability Distribution

Definition 1.13 The joint probability distribution Φ : <n → [0, 1] of a ran-dom vector (ξ1, ξ2, · · · , ξn) is defined by

Φ(x1, x2, · · · , xn) = Prω ∈ Ω

∣∣ ξ1(ω) ≤ x1, ξ2(ω) ≤ x2, · · · , ξn(ω) ≤ xn

.

Definition 1.14 The joint probability density function φ: <n → [0,+∞) ofa random vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn

holds for all (x1, x2, · · · , xn) ∈ <n, where Φ is the probability distribution ofthe random vector (ξ1, ξ2, · · · , ξn).

1.4 Independence

Definition 1.15 The random variables ξ1, ξ2, · · · , ξm are said to be indepen-dent if

Pr

m⋂

i=1

ξi ∈ Bi

=

m∏i=1

Prξi ∈ Bi (1.19)

for any Borel sets B1, B2, · · · , Bm of <.

Theorem 1.9 Let ξ1, ξ2, · · · , ξm be independent random variables, and f1,f2, · · · , fn are measurable functions. Then f1(ξ1), f2(ξ2), · · · , fm(ξm) are in-dependent random variables.

Proof: For any Borel sets of B1, B2, · · · , Bm of <, we have

Pr

m⋂

i=1

fi(ξi) ∈ Bi

= Pr

m⋂

i=1

ξi ∈ f−1i (Bi)

=m∏

i=1

Prξi ∈ f−1i (Bi) =

m∏i=1

Prfi(ξi) ∈ Bi.

Thus f1(ξ1), f2(ξ2), · · · , fm(ξm) are independent random variables.

Theorem 1.10 Let ξi be random variables with probability distributions Φi,i = 1, 2, · · · ,m, respectively, and Φ the probability distribution of the randomvector (ξ1, ξ2, · · · , ξm). Then ξ1, ξ2, · · · , ξm are independent if and only if

Φ(x1, x2, · · · , xm) = Φ1(x1)Φ2(x2) · · ·Φm(xm) (1.20)

for all (x1, x2, · · · , xm) ∈ <m.


Proof: If ξ1, ξ2, · · · , ξm are independent random variables, then we have

Φ(x1, x2, · · · , xm) = Prξ1 ≤ x1, ξ2 ≤ x2, · · · , ξm ≤ xm= Prξ1 ≤ x1Prξ2 ≤ x2 · · ·Prξm ≤ xm= Φ1(x1)Φ2(x2) · · ·Φm(xm)

for all (x1, x2, · · · , xm) ∈ <m.Conversely, assume that (1.20) holds. Let x2, x3, · · · , xm be fixed real

numbers, and C the class of all subsets C of < for which the relation

Prξ1 ∈ C, ξ2 ≤ x2, · · · , ξm ≤ xm = Prξ1 ∈ Cm∏

i=2

Prξi ≤ xi (1.21)

holds. We will show that C contains all Borel sets of <. It follows fromthe probability continuity theorem and relation (1.21) that C is a monotoneclass. It is also clear that C contains all intervals of the form (−∞, a], (a, b],(b,∞) and ∅. Let F be the algebra consisting of all finite unions of disjointsets of the form (−∞, a], (a, b], (b,∞) and ∅. Note that for any disjoint setsC1, C2, · · · , Ck of F and C = C1 ∪ C2 ∪ · · · ∪ Ck, we have

Prξ1 ∈ C, ξ2 ≤ x2, · · · , ξm ≤ xm

=m∑

j=1

Prξ1 ∈ Cj , ξ2 ≤ x2, · · · , ξm ≤ xm

= Prξ1 ∈ CPrξ2 ≤ x2 · · ·Prξm ≤ xm.


Applying the same reasoning to each ξi in turn, we obtain the indepen-dence of the random variables.

Theorem 1.11 Let ξi be random variables with probability density functionsφi, i = 1, 2, · · · ,m, respectively, and φ the probability density function of therandom vector (ξ1, ξ2, · · · , ξm). Then ξ1, ξ2, · · · , ξm are independent if andonly if

φ(x1, x2, · · · , xm) = φ1(x1)φ2(x2) · · ·φm(xm) (1.22)

for almost all (x1, x2, · · · , xm) ∈ <m.

Proof: If φ(x1, x2, · · · , xm) = φ1(x1)φ2(x2) · · ·φm(xm) a.e., then we have

Φ(x1, x2, · · · , xm) =∫ x1

−∞

∫ x2

−∞· · ·∫ xm

−∞φ(t1, t2, · · · , tm)dt1dt2 · · ·dtm

=∫ x1

−∞

∫ x2

−∞· · ·∫ xm

−∞φ1(t1)φ2(t2) · · ·φm(tm)dt1dt2 · · ·dtm

=∫ x1

−∞φ1(t1)dt1

∫ x2

−∞φ2(t2)dt2 · · ·

∫ xm

−∞φm(tm)dtm

= Φ1(x1)Φ2(x2) · · ·Φm(xm)

Section 1.5 - Identical Distribution 13

for all (x1, x2, · · · , xm) ∈ <m. Thus ξ1, ξ2, · · · , ξm are independent. Con-versely, if ξ1, ξ2, · · · , ξm are independent, then for any (x1, x2, · · · , xm) ∈ <m,we have Φ(x1, x2, · · · , xm) = Φ1(x1)Φ2(x2) · · ·Φm(xm). Hence

Φ(x1, x2, · · · , xm) =∫ x1

−∞

∫ x2

−∞· · ·∫ xm

−∞φ1(t1)φ2(t2) · · ·φm(tm)dt1dt2 · · ·dtm

which implies that φ(x1, x2, · · · , xm) = φ1(x1)φ2(x2) · · ·φm(xm) a.e.

Example 1.10: Let ξ1, ξ2, · · · , ξm be independent random variables withprobability density functions φ1, φ2, · · · , φm, respectively, and f : <m →< a measurable function. Then for any Borel set B of real numbers, theprobability Prf(ξ1, ξ2, · · · , ξm) ∈ B is∫ ∫

· · ·∫

f(x1,x2,··· ,xm)∈B

φ1(x1)φ2(x2) · · ·φm(xm)dx1dx2 · · ·dxm.

1.5 Identical Distribution

Definition 1.16 The random variables ξ and η are said to be identicallydistributed if

Prξ ∈ B = Prξ ∈ B (1.23)

for any Borel set B of <.

Theorem 1.12 The random variables ξ and η are identically distributed ifand only if they have the same probability distribution.

Proof: Let Φ and Ψ be the probability distributions of ξ and η, respectively.If ξ and η are identically distributed random variables, then, for any x ∈ <,we have

Φ(x) = Prξ ∈ (−∞, x] = Prη ∈ (−∞, x] = Ψ(x).

Thus ξ and η have the same probability distribution.Conversely, assume that ξ and η have the same probability distribution.

Let C be the class of all subsets C of < for which the relation

Prξ ∈ C = Prη ∈ C (1.24)

holds. We will show that C contains all Borel sets of <. It follows fromthe probability continuity theorem and relation (1.24) that C is a monotoneclass. It is also clear that C contains all intervals of the form (−∞, a], (a, b],(b,∞) and ∅ since ξ and η have the same probability distribution. Let F bethe algebra consisting of all finite unions of disjoint sets of the form (−∞, a],


(a, b], (b,∞) and ∅. Note that for any disjoint sets C1, C2, · · · , Ck of F andC = C1 ∪ C2 ∪ · · · ∪ Ck, we have

Prξ ∈ C =k∑

j=1

Prξ ∈ Cj =k∑

j=1

Prη ∈ Cj = Prη ∈ C.


Theorem 1.13 Let ξ and η be two random variables whose probability den-sity functions exist. Then ξ and η are identically distributed if and only ifthey have the same probability density function.

Proof: It follows from Theorem 1.12 that the random variables ξ and η areidentically distributed if and only if they have the same probability distribu-tion, if and only if they have the same probability density function.

1.6 Expected Value

Definition 1.17 Let ξ be a random variable. Then the expected value of ξis defined by

E[ξ] =∫ +∞

0

Prξ ≥ rdr −∫ 0

−∞Prξ ≤ rdr (1.25)

provided that at least one of the two integrals is finite.

Example 1.11: Let ξ be a uniformly distributed random variable U(a, b).If a ≥ 0, then Prξ ≤ r ≡ 0 when r ≤ 0, and

Prξ ≥ r =

1, if r ≤ a

(b− r)/(b− a), if a ≤ r ≤ b

0, if r ≥ b,

E[ξ] =

(∫ a

0

1dr +∫ b

a

b− r

b− adr +

∫ +∞

b

0dr

)−∫ 0

−∞0dr =

a+ b

2.

If b ≤ 0, then Prξ ≥ r ≡ 0 when r ≥ 0, and

Prξ ≤ r =

1, if r ≥ b

(r − a)/(b− a), if a ≤ r ≤ b

0, if r ≤ a,

Section 1.6 - Expected Value 15

E[ξ] =∫ +∞

0

0dr −

(∫ a

−∞0dr +

∫ b

a

r − a

b− adr +

∫ 0

b

1dr

)=a+ b

2.

If a < 0 < b, then

Prξ ≥ r =

(b− r)/(b− a), if 0 ≤ r ≤ b

0, if r ≥ b,

Prξ ≤ r =

0, if r ≤ a

(r − a)/(b− a), if a ≤ r ≤ 0,

E[ξ] =

(∫ b

0

b− r

b− adr +

∫ +∞

b

0dr

)−(∫ a

−∞0dr +

∫ 0

a

r − a

b− adr)

=a+ b

2.

Thus we always have the expected value (a+ b)/2.

Example 1.12: Let ξ be an exponentially distributed random variableEXP(β). Then its expected value is β.

Example 1.13: Let ξ be a normally distributed random variable N (µ, σ2).Then its expected value is µ.

Example 1.14: Assume that ξ is a discrete random variable taking valuesxi with probabilities pi, i = 1, 2, · · · ,m, respectively. It follows from thedefinition of expected value operator that

E[ξ] =m∑

i=1

pixi.

Theorem 1.14 Let ξ be a random variable whose probability density functionφ exists. If the Lebesgue integral

∫ +∞

−∞xφ(x)dx

is finite, then we have

E[ξ] =∫ +∞

−∞xφ(x)dx. (1.26)


Proof: It follows from Definition 1.17 and Fubini Theorem that

E[ξ] =∫ +∞

0

Prξ ≥ rdr −∫ 0

−∞Prξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.

The theorem is proved.

Theorem 1.15 Let ξ be a random variable with probability distribution Φ.If the Lebesgue-Stieltjes integral∫ +∞

−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (1.27)

Proof: Since the Lebesgue-Stieltjes integral∫ +∞−∞ xdΦ(x) is finite, we imme-

diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0

xdΦ(x), limy→−∞

∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y

xdΦ(x) = 0, limy→−∞

∫ y

−∞xdΦ(x) = 0.

It follows from∫ +∞

y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y(1− Φ(y)) ≥ 0, if y > 0,

∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, if y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


Let 0 = x0 < x1 < x2 < · · · < xn = y be a partition of [0, y]. Then we have

n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Prξ ≥ rdr

as max|xi+1 − xi| : i = 0, 1, · · · , n− 1 → 0. Since

n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) = y(Φ(y)− 1) → 0

as y → +∞. This fact implies that∫ +∞

0

Prξ ≥ rdr =∫ +∞

0

xdΦ(x).

A similar way may prove that

−∫ 0

−∞Prξ ≤ rdr =

∫ 0

−∞xdΦ(x).

Thus (1.27) is verified by the above two equations.

Linearity of Expected Value Operator

Theorem 1.16 Let ξ and η be random variables with finite expected values.Then for any numbers a and b, we have

E[aξ + bη] = aE[ξ] + bE[η]. (1.28)

Proof: Step 1: We first prove that E[ξ + b] = E[ξ] + b for any real numberb. When b ≥ 0, we have

E[ξ + b] =∫ ∞

0

Prξ + b ≥ rdr −∫ 0

−∞Prξ + b ≤ rdr

=∫ ∞

0

Prξ ≥ r − bdr −∫ 0

−∞Prξ ≤ r − bdr

= E[ξ] +∫ b

0

(Prξ ≥ r − b+ Prξ < r − b) dr

= E[ξ] + b.

If b < 0, then we have

E[ξ + b] = E[ξ]−∫ 0

b

(Prξ ≥ r − b+ Prξ < r − b) dr = E[ξ] + b.


Step 2: We prove that E[aξ] = aE[ξ] for any real number a. If a = 0,then the equation E[aξ] = aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ ∞

0

Praξ ≥ rdr −∫ 0

−∞Praξ ≤ rdr

=∫ ∞

0

Prξ ≥ r

a

dr −

∫ 0

−∞Prξ ≤ r

a

dr

= a

∫ ∞

0

Prξ ≥ r

a

d( ra

)− a

∫ 0

−∞Prξ ≤ r

a

d( ra

)= aE[ξ].

If a < 0, we have

E[aξ] =∫ ∞

0

Praξ ≥ rdr −∫ 0

−∞Praξ ≤ rdr

=∫ ∞

0

Prξ ≤ r

a

dr −

∫ 0

−∞Prξ ≥ r

a

dr

= a

∫ ∞

0

Prξ ≥ r

a

d( ra

)− a

∫ 0

−∞Prξ ≤ r

a

d( ra

)= aE[ξ].

Step 3: We prove that E[ξ + η] = E[ξ] + E[η] when both ξ and ηare nonnegative simple random variables taking values a1, a2, · · · , am andb1, b2, · · · , bn, respectively. Then ξ + η is also a nonnegative simple randomvariable taking values ai + bj , i = 1, 2, · · · ,m, j = 1, 2, · · · , n. Thus we have

E[ξ + η] =m∑

i=1

n∑j=1

(ai + bj) Prξ = ai, η = bj

=m∑

i=1

n∑j=1

ai Prξ = ai, η = bj+m∑

i=1

n∑j=1

bj Prξ = ai, η = bj

=m∑

i=1

ai Prξ = ai+n∑

j=1

bj Prη = bj

= E[ξ] + E[η].

Step 4: We prove that E[ξ + η] = E[ξ] + E[η] when both ξ and η arenonnegative random variables. For every i ≥ 1 and every ω ∈ Ω, we define

ξi(ω) =

k − 1

2i, if

k − 12i

≤ ξ(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ ξ(ω),


ηi(ω) =

k − 1

2i, if

k − 12i

≤ η(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ η(ω).

Then ξi, ηi and ξi + ηi are three sequences of nonnegative simplerandom variables such that ξi ↑ ξ, ηi ↑ η and ξi + ηi ↑ ξ + η as i→∞. Notethat the functions Prξi > r, Prηi > r, Prξi + ηi > r, i = 1, 2, · · · arealso simple. It follows from the probability continuity theorem that

Prξi > r ↑ Prξ > r, ∀r ≥ 0

as i→∞. Since the expected value E[ξ] exists, we have

E[ξi] =∫ +∞

0

Prξi > rdr →∫ +∞

0

Prξ > rdr = E[ξ]

as i→∞. Similarly, we may prove that E[ηi] → E[η] and E[ξi+ηi] → E[ξ+η]as i→∞. It follows from Step 3 that E[ξ + η] = E[ξ] + E[η].

Step 5: We prove that E[ξ+η] = E[ξ]+E[η] when ξ and η are arbitraryrandom variables. Define

ξi(ω) =

ξ(ω), if ξ(ω) ≥ −i−i, otherwise,

ηi(ω) =

η(ω), if η(ω) ≥ −i−i, otherwise.

Since the expected values E[ξ] and E[η] are finite, we have

limi→∞

E[ξi] = E[ξ], limi→∞

E[ηi] = E[η], limi→∞

E[ξi + ηi] = E[ξ + η].

Note that (ξi + i) and (ηi + i) are nonnegative random variables. It followsfrom Steps 1 and 4 that

E[ξ + η] = limi→∞

E[ξi + ηi]

= limi→∞

(E[(ξi + i) + (ηi + i)]− 2i)

= limi→∞

(E[ξi + i] + E[ηi + i]− 2i)

= limi→∞

(E[ξi] + i+ E[ηi] + i− 2i)

= limi→∞

E[ξi] + limi→∞

E[ηi]

= E[ξ] + E[η].

Step 6: The linearity E[aξ + bη] = aE[ξ] + bE[η] follows immediatelyfrom Steps 2 and 5. The theorem is proved.


Product of Independent Random Variables

Theorem 1.17 Let ξ and η be independent random variables with finite ex-pected values. Then the expected value of ξη exists and

E[ξη] = E[ξ]E[η]. (1.29)

Proof: Step 1: We first prove the case where both ξ and η are nonnega-tive simple random variables taking values a1, a2, · · · , am and b1, b2, · · · , bn,respectively. Then ξη is also a nonnegative simple random variable takingvalues aibj , i = 1, 2, · · · ,m, j = 1, 2, · · · , n. It follows from the independenceof ξ and η that

E[ξη] =m∑

i=1

n∑j=1

aibj Prξ = ai, η = bj

=m∑

i=1

n∑j=1

aibj Prξ = aiPrη = bj

=(

m∑i=1

ai Prξ = ai)(

n∑j=1

bj Prη = bj

)= E[ξ]E[η].

Step 2: Next we prove the case where ξ and η are nonnegative randomvariables. For every i ≥ 1 and every ω ∈ Ω, we define

ξi(ω) =

k − 1

2i, if

k − 12i

≤ ξ(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ ξ(ω),

ηi(ω) =

k − 1

2i, if

k − 12i

≤ η(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ η(ω).

Then ξi, ηi and ξiηi are three sequences of nonnegative simple randomvariables such that ξi ↑ ξ, ηi ↑ η and ξiηi ↑ ξη as i → ∞. It follows fromthe independence of ξ and η that ξi and ηi are independent. Hence we haveE[ξiηi] = E[ξi]E[ηi] for i = 1, 2, · · · It follows from the probability continuitytheorem that Prξi > r, i = 1, 2, · · · are simple functions such that

Prξi > r ↑ Prξ > r, for all r ≥ 0

as i→∞. Since the expected value E[ξ] exists, we have

E[ξi] =∫ +∞

0

Prξi > rdr →∫ +∞

0

Prξ > rdr = E[ξ]

as i → ∞. Similarly, we may prove that E[ηi] → E[η] and E[ξiηi] → E[ξη]as i→∞. Therefore E[ξη] = E[ξ]E[η].


Step 3: Finally, if ξ and η are arbitrary independent random variables,then the nonnegative random variables ξ+ and η+ are independent and soare ξ+ and η−, ξ− and η+, ξ− and η−. Thus we have

E[ξ+η+] = E[ξ+]E[η+], E[ξ+η−] = E[ξ+]E[η−],

E[ξ−η+] = E[ξ−]E[η+], E[ξ−η−] = E[ξ−]E[η−].

It follows thatE[ξη] = E[(ξ+ − ξ−)(η+ − η−)]

= E[ξ+η+]− E[ξ+η−]− E[ξ−η+] + E[ξ−η−]

= E[ξ+]E[η+]− E[ξ+]E[η−]− E[ξ−]E[η+] + E[ξ−]E[η−]

= (E[ξ+]− E[ξ−]) (E[η+]− E[η−])

= E[ξ+ − ξ−]E[η+ − η−]

= E[ξ]E[η]

which proves the theorem.

Expected Value of Function of Random Variable

Theorem 1.18 Let ξ be a random variable with probability distribution Φ,and f : < → < a measurable function. If the Lebesgue-Stieltjes integral∫ +∞

−∞f(x)dΦ(x)


E[f(ξ)] =∫ +∞

−∞f(x)dΦ(x). (1.30)

Proof: It follows from the definition of expected value operator that

E[f(ξ)] =∫ +∞

0

Prf(ξ) ≥ rdr −∫ 0

−∞Prf(ξ) ≤ rdr. (1.31)

If f is a nonnegative simple measurable function, i.e.,

f(x) =

a1, if x ∈ B1

a2, if x ∈ B2

· · ·am, if x ∈ Bm

where B1, B2, · · · , Bm are mutually disjoint Borel sets, then we have

E[f(ξ)] =∫ +∞

0

Prf(ξ) ≥ rdr =m∑

i=1

ai Prξ ∈ Bi

=m∑

i=1

ai

∫Bi

dΦ(x) =∫ +∞

−∞f(x)dΦ(x).


We next prove the case where f is a nonnegative measurable function. Letf1, f2, · · · be a sequence of nonnegative simple functions such that fi ↑ f asi→∞. We have proved that

E[fi(ξ)] =∫ +∞

0

Prfi(ξ) ≥ rdr =∫ +∞

−∞fi(x)dΦ(x).

In addition, it is also known that Prfi(ξ) > r ↑ Prf(ξ) > r as i→∞ forr ≥ 0. It follows from the monotone convergence theorem that

E[f(ξ)] =∫ +∞

0

Prf(ξ) > rdr

= limi→∞

∫ +∞

0

Prfi(ξ) > rdr

= limi→∞

∫ +∞

−∞fi(x)dΦ(x)

=∫ +∞

−∞f(x)dΦ(x).

Finally, if f is an arbitrary measurable function, then we have f = f+ − f−

andE[f(ξ)] = E[f+(ξ)− f−(ξ)] = E[f+(ξ)]− E[f−(ξ)]

=∫ +∞

−∞f+(x)dΦ(x)−

∫ +∞

−∞f−(x)dΦ(x)

=∫ +∞

−∞f(x)dΦ(x).


Sum of a Random Number of Random Variables

Theorem 1.19 (Wald Identity) Assume that ξi is a sequence of iid ran-dom variables, and η is a positive random integer (i.e., a random variabletaking “positive integer” values) that is independent of the sequence ξi.Then we have

E

[η∑

i=1

ξi

]= E[η]E[ξ1]. (1.32)

Proof: Since η is independent of the sequence ξi, we have

Pr

η∑

i=1

ξi ≥ r

=

∞∑k=1

Prη = kPr ξ1 + ξ2 + · · ·+ ξk ≥ r .

Section 1.7 - Variance 23

If ξi are nonnegative random variables, then we have

E

[η∑

i=1

ξi

]=∫ +∞

0

Pr

η∑

i=1

ξi ≥ r

dr

=∫ +∞

0

∞∑k=1

Prη = kPr ξ1 + ξ2 + · · ·+ ξk ≥ rdr

=∞∑

k=1

Prη = k∫ +∞

0

Pr ξ1 + ξ2 + · · ·+ ξk ≥ rdr

=∞∑

k=1

Prη = k (E[ξ1] + E[ξ2] + · · ·+ E[ξk])

=∞∑

k=1

Prη = kkE[ξ1] (by iid hypothesis)

= E[η]E[ξ1].

If ξi are arbitrary random variables, then ξi = ξ+i − ξ−i , and

E

[η∑

i=1

ξi

]= E

[η∑

i=1

(ξ+i − ξ−i )

]= E

[η∑

i=1

ξ+i −η∑

i=1

ξ−i

]

= E

[η∑

i=1

ξ+i

]− E

[η∑

i=1

ξ−i

]= E[η]E[ξ+1 ]− E[η]E[ξ−1 ]

= E[η](E[ξ+1 ]− E[ξ−1 ]) = E[η]E[ξ+1 − ξ−1 ] = E[η]E[ξ1].

The theorem is thus proved.

1.7 Variance

Definition 1.18 Let ξ be a random variable with finite expected value e.Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

The variance of a random variable provides a measure of the spread of thedistribution around its expected value. A small value of variance indicatesthat the random variable is tightly concentrated around its expected value;and a large value of variance indicates that the random variable has a widespread around its expected value.

Example 1.15: Let ξ be a uniformly distributed random variable U(a, b).Then its expected value is (a+ b)/2 and variance is (b− a)2/12.

Example 1.16: Let ξ be an exponentially distributed random variableEXP(β). Then its expected value is β and variance is β2.


Example 1.17: Let ξ be a normally distributed random variable N (µ, σ2).Then its expected value is µ and variance is σ2.

Theorem 1.20 If ξ is a random variable whose variance exists, a and b arereal numbers, then V [aξ + b] = a2V [ξ].

Proof: It follows from the definition of variance that

V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 1.21 Let ξ be a random variable with expected value e. ThenV [ξ] = 0 if and only if Prξ = e = 1.

Proof: If V [ξ] = 0, then E[(ξ − e)2] = 0. Thus we have∫ +∞

0

Pr(ξ − e)2 ≥ rdr = 0

which implies Pr(ξ−e)2 ≥ r = 0 for any r > 0. Hence we have Pr(ξ−e)2 =0 = 1, i.e., Prξ = e = 1.

Conversely, if Prξ = e = 1, then we have Pr(ξ − e)2 = 0 = 1 andPr(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

Pr(ξ − e)2 ≥ rdr = 0.

Theorem 1.22 If ξ1, ξ2, · · · , ξn are independent random variables with finiteexpected values, then

V [ξ1 + ξ2 + · · ·+ ξn] = V [ξ1] + V [ξ2] + · · ·+ V [ξn]. (1.33)


V

[n∑

i=1

ξi

]= E

[(ξ1 + ξ2 + · · ·+ ξn − E[ξ1]− E[ξ2]− · · · − E[ξn])2

]=

n∑i=1

E[(ξi − E[ξi])2

]+ 2

n−1∑i=1

n∑j=i+1

E [(ξi − E[ξi])(ξj − E[ξj ])] .

Since ξ1, ξ2, · · · , ξn are independent, E [(ξi − E[ξi])(ξj − E[ξj ])] = 0 for alli, j with i 6= j. Thus (1.33) holds.

Maximum Variance Theorem

Let ξ be a random variable that takes values in [a, b], but whose probabilitydistribution is otherwise arbitrary. If its expected value is given, what is thepossible maximum variance? The maximum variance theorem will answerthis question, thus playing an important role in treating games against nature.

Section 1.8 - Moments 25

Theorem 1.23 (Edmundson-Madansky Inequality) Let f be a convex func-tion on [a, b], and ξ a random variable that takes values in [a, b] and hasexpected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (1.34)

Proof: For each ω ∈ Ω, we have a ≤ ξ(ω) ≤ b and

ξ(ω) =b− ξ(ω)b− a

a+ξ(ω)− a

b− ab.

It follows from the convexity of f that

f(ξ(ω)) ≤ b− ξ(ω)b− a

f(a) +ξ(ω)− a

b− af(b).

Taking expected values on both sides, we obtain (1.34).

Theorem 1.24 (Maximum Variance Theorem) Let ξ be a random variablethat takes values in [a, b] and has expected value e. Then

V [ξ] ≤ (e− a)(b− e) (1.35)

and equality holds if the random variable ξ is determined by

Prξ = x =

b− e

b− a, if x = a

e− a

b− a, if x = b.

(1.36)

Proof: It follows from Theorem 1.23 immediately by defining f(x) = (x−e)2.It is also easy to verify that the random variable determined by (1.36) hasvariance (e− a)(b− e). The theorem is proved.

1.8 Moments

Definition 1.19 Let ξ be a random variable, and k a positive number. Then(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.

Note that the first central moment is always 0, the first moment is justthe expected value, and the second central moment is just the variance.

Theorem 1.25 Let ξ be a nonnegative random variable, and k a positivenumber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1 Prξ ≥ rdr. (1.37)


Proof: It follows from the nonnegativity of ξ that

E[ξk] =∫ ∞

0

Prξk ≥ xdx =∫ ∞

0

Prξ ≥ rdrk = k

∫ ∞

0

rk−1 Prξ ≥ rdr.


Theorem 1.26 Let ξ be a random variable that takes values in [a, b] and hasexpected value e. Then for any positive integer k, the kth absolute momentand kth absolute central moment satisfy the following inequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (1.38)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (1.39)

Proof: It follows from Theorem 1.23 immediately by defining f(x) = |x|kand f(x) = |x− e|k.

1.9 Critical Values

Let ξ be a random variable. In order to measure it, we may use its expectedvalue. Alternately, we may employ α-optimistic value and α-pessimistic valueas a ranking measure.

Definition 1.20 Let ξ be a random variable, and α ∈ (0, 1]. Then

ξsup(α) = supr∣∣ Pr ξ ≥ r ≥ α

(1.40)

is called the α-optimistic value of ξ, and

ξinf(α) = infr∣∣ Pr ξ ≤ r ≥ α

(1.41)

is called the α-pessimistic value of ξ.

This means that the random variable ξ will reach upwards of the α-optimistic value ξsup(α) at least α of time, and will be below the α-pessimisticvalue ξinf(α) at least α of time. The optimistic value is also called percentile.

Theorem 1.27 Let ξ be a random variable. Then we have

Prξ ≥ ξsup(α) ≥ α, Prξ ≤ ξinf(α) ≥ α (1.42)

where ξsup(α) and ξinf(α) are the α-optimistic and α-pessimistic values of therandom variable ξ, respectively.

Section 1.9 - Critical Values 27

Proof: It follows from the definition of the optimistic value that there existsan increasing sequence ri such that Prξ ≥ ri ≥ α and ri ↑ ξsup(α) asi → ∞. Since ω|ξ(ω) ≥ ri ↓ ω|ξ(ω) ≥ ξsup(α), it follows from theprobability continuity theorem that

Prξ ≥ ξsup(α) = limi→∞

Prξ ≥ ri ≥ α.

The inequality Prξ ≤ ξinf(α) ≥ α may be proved similarly.

Example 1.18: Note that Prξ ≥ ξsup(α) > α and Prξ ≤ ξinf(α) > αmay hold. For example,

ξ =

0 with probability 0.41 with probability 0.6.

If α = 0.8, then ξsup(0.8) = 0 which makes Prξ ≥ ξsup(0.8) = 1 > 0.8. Inaddition, ξinf(0.8) = 1 and Prξ ≤ ξinf(0.8) = 1 > 0.8.

Theorem 1.28 Let ξ be a random variable. Then we have(a) ξinf(α) is an increasing and left-continuous function of α;(b) ξsup(α) is a decreasing and left-continuous function of α.

Proof: (a) It is easy to prove that ξinf(α) is an increasing function of α.Next, we prove the left-continuity of ξinf(α) with respect to α. Let αi bean arbitrary sequence of positive numbers such that αi ↑ α. Then ξinf(αi)is an increasing sequence. If the limitation is equal to ξinf(α), then the left-continuity is proved. Otherwise, there exists a number z∗ such that

limi→∞

ξinf(αi) < z∗ < ξinf(α).

Thus Prξ ≤ z∗ ≥ αi for each i. Letting i → ∞, we get Prξ ≤ z∗ ≥ α.Hence z∗ ≥ ξinf(α). A contradiction proves the left-continuity of ξinf(α) withrespect to α. The part (b) may be proved similarly.

Theorem 1.29 Let ξ be a random variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).

Proof: Part (a): Write ξ(α) = (ξinf(α) + ξsup(α))/2. If ξinf(α) < ξsup(α),then we have

1 ≥ Prξ < ξ(α)+ Prξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α). Part (b): Assume that ξinf(α) >ξsup(α). It follows from the definition of ξinf(α) that Prξ ≤ ξ(α) < α.Similarly, it follows from the definition of ξsup(α) that Prξ ≥ ξ(α) < α.Thus

1 ≤ Prξ ≤ ξ(α)+ Prξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is proved.


Theorem 1.30 Let ξ be a random variable. Then we have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Proof: (a) If c = 0, then it is obviously valid. When c > 0, we have

(cξ)sup(α) = sup r | Prcξ ≥ r ≥ α= c sup r/c | Pr ξ ≥ r/c ≥ α= cξsup(α).

A similar way may prove that (cξ)inf(α) = cξinf(α).(b) In order to prove this part, it suffices to verify that (−ξ)sup(α) =

−ξinf(α) and (−ξ)inf(α) = −ξsup(α). In fact, for any α ∈ (0, 1], we have

(−ξ)sup(α) = supr | Pr−ξ ≥ r ≥ α= − inf−r | Prξ ≤ −r ≥ α= −ξinf(α).

Similarly, we may prove that (−ξ)inf(α) = −ξsup(α). The theorem is proved.

1.10 Entropy

Given a random variable, what is the degree of difficulty of predicting thespecified value that the random variable will take? In order to answer thisquestion, Shannon [207] defined a concept of entropy as a measure of uncer-tainty.

Entropy of Discrete Random Variables

Definition 1.21 Let ξ be a discrete random variable taking values xi withprobabilities pi, i = 1, 2, · · · , respectively. Then its entropy is defined by

H[ξ] = −∞∑

i=1

pi ln pi. (1.43)

It should be noticed that the entropy depends only on the number ofvalues and their probabilities and does not depend on the actual values thatthe random variable takes.

Theorem 1.31 Let ξ be a discrete random variable taking values xi withprobabilities pi, i = 1, 2, · · · , respectively. Then

H[ξ] ≥ 0 (1.44)

and equality holds if and only if there exists an index k such that pk = 1, i.e.,ξ is essentially a deterministic number.

Section 1.10 - Entropy 29

................................................................................................................................................................................................................................................................................................................................... ...............

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

...................

...............

0 1t

S(t)

.....................................................................................

..................................................................................................................................................................................................................................................................................................................

Figure 1.1: Function S(t) = −t ln t is concave

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifpi = 0 or 1 for each i. That is, there exists one and only one index k suchthat pk = 1. The theorem is proved.

This theorem states that the entropy of a discrete random variable reachesits minimum 0 when the random variable degenerates to a deterministic num-ber. In this case, there is no uncertainty.

Theorem 1.32 Let ξ be a simple random variable taking values xi with prob-abilities pi, i = 1, 2, · · · , n, respectively. Then

H[ξ] ≤ lnn (1.45)

and equality holds if and only if pi ≡ 1/n for all i = 1, 2, · · · , n.

Proof: Since the function S(t) is a concave function of t and p1 + p2 + · · ·+pn = 1, we have

−n∑

i=1

pi ln pi ≤ −n

(1n

n∑i=1

pi

)ln

(1n

n∑i=1

pi

)= lnn

which implies that H[ξ] ≤ lnn and equality holds if and only if p1 = p2 =· · · = pn, i.e., pi ≡ 1/n for all i = 1, 2, · · · , n.

This theorem states that the entropy of a simple random variable reachesits maximum lnn when all outcomes are equiprobable. In this case, there isno preference among all the values that the random variable will take.

Entropy of Absolutely Continuous Random Variables

Definition 1.22 Let ξ be a random variable with probability density functionφ. Then its entropy is defined by

H[ξ] = −∫ +∞

−∞φ(x) lnφ(x)dx. (1.46)


Example 1.19: Let ξ be a uniformly distributed random variable on [a, b].Then its entropy is H[ξ] = ln(b− a). This example shows that the entropy ofabsolutely continuous random variable may assume both positive and nega-tive values since ln(b− a) < 0 if b− a < 1; and ln(b− a) > 0 if b− a > 1.

Example 1.20: Let ξ be an exponentially distributed random variable withexpected value β. Then its entropy is H[ξ] = 1 + lnβ.

Example 1.21: Let ξ be a normally distributed random variable with ex-pected value e and variance σ2. Then its entropy is H[ξ] = 1/2 + ln

√2πσ.

Theorem 1.33 Let ξ be an absolutely continuous random variable. ThenH[aξ + b] = |a|H[ξ] for any real numbers a and b.

Proof: It follows immediately from the definition.

Maximum Entropy Principle

Given some constraints, for example, expected value and variance, there areusually multiple compatible probability distributions. For this case, we wouldlike to select the distribution that maximizes the value of entropy and satisfiesthe prescribed constraints. This method is often referred to as the maximumentropy principle (Jaynes [67]).

Example 1.22: Let ξ be an absolutely continuous random variable on [a, b].The maximum entropy principle attempts to find the probability densityfunction φ(x) that maximizes the entropy

−∫ b

a

φ(x) lnφ(x)dx

subject to the natural constraint∫ b

aφ(x)dx = 1. The Lagrangian is

L = −∫ b

a

φ(x) lnφ(x)dx− λ

(∫ b

a

φ(x)dx− 1

).

It follows from the Euler-Lagrange equation that the maximum entropy prob-ability density function meets

lnφ(x) + 1 + λ = 0

and has the form φ(x) = exp(−1 − λ). Substituting it into the naturalconstraint, we get

φ∗(x) =1

b− a, a ≤ x ≤ b

which is just the uniformly distributed random variable, and the maximumentropy is H[ξ∗] = ln(b− a).


Example 1.23: Let ξ be an absolutely continuous random variable on [0,∞).Assume that the expected value of ξ is prescribed to be β. The maximumentropy probability density function φ(x) should maximize the entropy

−∫ +∞

0

φ(x) lnφ(x)dx

subject to the constraints∫ +∞

0

φ(x)dx = 1,∫ +∞

0

xφ(x)dx = β.

The Lagrangian is

L = −∫ ∞

0

φ(x) lnφ(x)dx−λ1

(∫ ∞

0

φ(x)dx− 1)−λ2

(∫ ∞

0

xφ(x)dx− β

).

The maximum entropy probability density function meets Euler-Lagrangeequation

lnφ(x) + 1 + λ1 + λ2x = 0

and has the form φ(x) = exp(−1 − λ1 − λ2x). Substituting it into theconstraints, we get

φ∗(x) =1β

exp(−xβ

), x ≥ 0

which is just the exponentially distributed random variable, and the maxi-mum entropy is H[ξ∗] = 1 + lnβ.

Example 1.24: Let ξ be an absolutely continuous random variable on(−∞,+∞). Assume that the expected value and variance of ξ are prescribedto be µ and σ2, respectively. The maximum entropy probability densityfunction φ(x) should maximize the entropy

−∫ +∞

−∞φ(x) lnφ(x)dx

subject to the constraints∫ +∞

−∞φ(x)dx = 1,

∫ +∞

−∞xφ(x)dx = µ,

∫ +∞

−∞(x− µ)2φ(x)dx = σ2.

The Lagrangian is

L = −∫ +∞

−∞φ(x) lnφ(x)dx− λ1

(∫ +∞

−∞φ(x)dx− 1

)

−λ2

(∫ +∞

−∞xφ(x)dx− µ

)− λ3

(∫ +∞

−∞(x− µ)2φ(x)dx− σ2

).


The maximum entropy probability density function meets Euler-Lagrangeequation

lnφ(x) + 1 + λ1 + λ2x+ λ3(x− µ)2 = 0

and has the form φ(x) = exp(−1 − λ1 − λ2x − λ3(x − µ)2). Substituting itinto the constraints, we get

φ∗(x) =1

σ√

2πexp

(− (x− µ)2

2σ2

), x ∈ <

which is just the normally distributed random variable, and the maximumentropy is H[ξ∗] = 1/2 + ln

√2πσ.

1.11 Distance

Distance is a powerful concept in many disciplines of science and engineering.This section introduces the distance between random variables.

Definition 1.23 The distance between random variables ξ and η is definedas

d(ξ, η) = E[|ξ − η|]. (1.47)

Theorem 1.34 Let ξ, η, τ be random variables, and let d(·, ·) be the distance.Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ d(ξ, τ) + d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.The part (d) is proved by the following relation,

E[|ξ − η|] ≤ E[|ξ − τ |+ |η − τ |] = E[|ξ − τ |] + E[|η − τ |].

1.12 Inequalities

It is well-known that there are several inequalities in probability theory, suchas Markov inequality, Chebyshev inequality, Holder’s inequality, Minkowskiinequality, and Jensen’s inequality. They play an important role in boththeory and applications.

Theorem 1.35 Let ξ be a random variable, and f a nonnegative measurablefunction. If f is even (i.e., f(x) = f(−x) for any x ∈ <) and increasing on[0,∞), then for any given number t > 0, we have

Pr|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (1.48)

Section 1.12 - Inequalities 33

Proof: It is clear that Pr|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Prf(ξ) ≥ rdr

=∫ +∞

0

Pr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

Pr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr · Pr|ξ| ≥ f−1(f(t))

= f(t) · Pr|ξ| ≥ t

which proves the inequality.

Theorem 1.36 (Markov Inequality) Let ξ be a random variable. Then forany given numbers t > 0 and p > 0, we have

Pr|ξ| ≥ t ≤ E[|ξ|p]tp

. (1.49)

Proof: It is a special case of Theorem 1.35 when f(x) = |x|p.

Example 1.25: For any given positive number t, we define a random variableas follows,

ξ =

0 with probability 1/2t with probability 1/2.

Then Prξ ≥ t = 1/2 = E[|ξ|p]/tp.

Theorem 1.37 (Chebyshev Inequality) Let ξ be a random variable whosevariance V [ξ] exists. Then for any given number t > 0, we have

Pr |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (1.50)

Proof: It is a special case of Theorem 1.35 when the random variable ξ isreplaced with ξ − E[ξ] and f(x) = x2.

Example 1.26: For any given positive number t, we define a random variableas follows,

ξ =

−t with probability 1/2t with probability 1/2.

Then Pr|ξ − E[ξ]| ≥ t = 1 = V [ξ]/t2.


Theorem 1.38 (Holder’s Inequality) Let p and q be two positive real num-bers with 1/p+1/q = 1, and let ξ and η be random variables with E[|ξ|p] <∞and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (1.51)

Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Nowwe assume E[|ξ|p] > 0 and E[|η|q] > 0. It is easy to prove that the functionf(x, y) = p

√x q√y is a concave function on D = (x, y) : x ≥ 0, y ≥ 0. Thus

for any point (x0, y0) with x0 > 0 and y0 > 0, there exist two real numbersa and b such that

f(x, y)− f(x0, y0) ≤ a(x− x0) + b(y − y0), ∀(x, y) ∈ D.

Letting x0 = E[|ξ|p], y0 = E[|η|q], x = |ξ|p and y = |η|q, we have

f(|ξ|p, |η|q)− f(E[|ξ|p], E[|η|q]) ≤ a(|ξ|p − E[|ξ|p]) + b(|η|q − E[|η|q]).

Taking the expected values on both sides, we obtain

E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).

Hence the inequality (1.51) holds.

Theorem 1.39 (Minkowski Inequality) Let p be a real number with p ≥ 1,and let ξ and η be random variables with E[|ξ|p] <∞ and E[|η|p] <∞. Thenwe have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (1.52)

Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Nowwe assume E[|ξ|p] > 0 and E[|η|p] > 0. It is easy to prove that the functionf(x, y) = ( p

√x + p

√y)p is a concave function on D = (x, y) : x ≥ 0, y ≥ 0.

Thus for any point (x0, y0) with x0 > 0 and y0 > 0, there exist two realnumbers a and b such that


Letting x0 = E[|ξ|p], y0 = E[|η|p], x = |ξ|p and y = |η|p, we have

f(|ξ|p, |η|p)− f(E[|ξ|p], E[|η|p]) ≤ a(|ξ|p − E[|ξ|p]) + b(|η|p − E[|η|p]).


E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Section 1.13 - Convergence Concepts 35

Theorem 1.40 (Jensen’s Inequality) Let ξ be a random variable, and f :< → < a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (1.53)

Especially, when f(x) = |x|p and p ≥ 1, we have |E[ξ]|p ≤ E[|ξ|p].

Proof: Since f is a convex function, for each y, there exists a number k suchthat f(x)− f(y) ≥ k · (x− y). Replacing x with ξ and y with E[ξ], we obtain

f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).

Taking the expected values on both sides, we have

E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0


1.13 Convergence Concepts

There are four main types of convergence concepts of random sequence:convergence almost surely (a.s.), convergence in probability, convergence inmean, and convergence in distribution.

Table 1.1: Relations among Convergence Concepts

Convergence Almost Surely Convergence → Convergence in Probability in Distribution

Convergence in Mean

Definition 1.24 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). The sequence ξi is said to be convergenta.s. to ξ if and only if there exists a set A ∈ A with PrA = 1 such that

limi→∞

|ξi(ω)− ξ(ω)| = 0 (1.54)

for every ω ∈ A. In that case we write ξi → ξ, a.s.

Definition 1.25 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). We say that the sequence ξi converges inprobability to ξ if

limi→∞

Pr |ξi − ξ| ≥ ε = 0 (1.55)

for every ε > 0.


Definition 1.26 Suppose that ξ, ξ1, ξ2, · · · are random variables with finiteexpected values on the probability space (Ω,A,Pr). We say that the sequenceξi converges in mean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (1.56)

In addition, the sequence ξi is said to converge in mean square to ξ if

limi→∞

E[|ξi − ξ|2] = 0. (1.57)

Definition 1.27 Suppose that Φ,Φ1,Φ2, · · · are the probability distributionsof random variables ξ, ξ1, ξ2, · · · , respectively. We say that ξi converges indistribution to ξ if Φi → Φ at any continuity point of Φ.

Convergence Almost Surely vs. Convergence in Probability

Theorem 1.41 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). Then ξi converges a.s. to ξ if and only if,for every ε > 0, we have

limn→∞

Pr

∞⋃i=n

|ξi − ξ| ≥ ε

= 0. (1.58)

Proof: For every i ≥ 1 and ε > 0, we define

X =ω ∈ Ω

∣∣ limi→∞

ξi(ω) 6= ξ(ω),

Xi(ε) =ω ∈ Ω

∣∣ |ξi(ω)− ξ(ω)| ≥ ε.

It is clear that

X =⋃ε>0

( ∞⋂n=1

∞⋃i=n

Xi(ε)

).

Note that ξi → ξ, a.s. if and only if PrX = 0. That is, ξi → ξ, a.s. if andonly if

Pr

∞⋂n=1

∞⋃i=n

Xi(ε)

= 0

for every ε > 0. Since∞⋃

i=n

Xi(ε) ↓∞⋂

n=1

∞⋃i=n

Xi(ε),

it follows from the probability continuity theorem that

limn→∞

Pr

∞⋃i=n

Xi(ε)

= Pr

∞⋂n=1

∞⋃i=n

Xi(ε)

= 0.



Theorem 1.42 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). If ξi converges a.s. to ξ, then ξi convergesin probability to ξ.

Proof: It follows from the convergence a.s. and Theorem 1.41 that

limn→∞

Pr

∞⋃i=n

|ξi − ξ| ≥ ε

= 0

for each ε > 0. For every n ≥ 1, since

|ξn − ξ| ≥ ε ⊂∞⋃

i=n

|ξi − ξ| ≥ ε,

we have Pr|ξn − ξ| ≥ ε → 0 as n→∞. Hence the theorem holds.

Example 1.27: Convergence in probability does not imply convergence a.s.For example, take (Ω,A,Pr) to be the interval [0, 1] with Borel algebra andLebesgue measure. For any positive integer i, there is an integer j such thati = 2j + k, where k is an integer between 0 and 2j − 1. We define a randomvariable on Ω by

ξi(ω) =

1, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ = 0. For any small number ε > 0, we have

Pr |ξi − ξ| ≥ ε =12j→ 0

as i→∞. That is, the sequence ξi converges in probability to ξ. However,for any ω ∈ [0, 1], there is an infinite number of intervals of the form [k/2j , (k+1)/2j ] containing ω. Thus ξi(ω) 6→ 0 as i→∞. In other words, the sequenceξi does not converge a.s. to ξ.

Convergence in Probability vs. Convergence in Mean

Theorem 1.43 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). If the sequence ξi converges in mean to ξ,then ξi converges in probability to ξ.

Proof: It follows from the Markov inequality that, for any given numberε > 0,

Pr |ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in probability to ξ.


Example 1.28: Convergence in probability does not imply convergence inmean. For example, take (Ω,A,Pr) to be ω1, ω2, · · · with Prωj = 1/2j

for j = 1, 2, · · · The random variables are defined by

ξiωj =

2i, if j = i0, otherwise


Pr |ξi − ξ| ≥ ε =12i→ 0.

That is, the sequence ξi converges in probability to ξ. However, we have

E [|ξi − ξ|] = 2i · 12i

= 1.

That is, the sequence ξi does not converge in mean to ξ.

Convergence Almost Surely vs. Convergence in Mean

Example 1.29: Convergence a.s. does not imply convergence in mean. Forexample, take (Ω,A,Pr) to be ω1, ω2, · · · with Prωj = 1/2j for j =1, 2, · · · The random variables are defined by

ξiωj =

2i, if j = i0, otherwise

for i = 1, 2, · · · and ξ = 0. Then ξi converges a.s. to ξ. However, thesequence ξi does not converge in mean to ξ.

Example 1.30: Convergence in mean does not imply convergence a.s. Forexample, take (Ω,A,Pr) to be the interval [0, 1] with Borel algebra andLebesgue measure. For any positive integer i, there is an integer j suchthat i = 2j + k, where k is an integer between 0 and 2j − 1. We define arandom variable on Ω by

ξi(ω) =

1, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ = 0. Then

E [|ξi − ξ|] =12j→ 0.

That is, the sequence ξi converges in mean to ξ. However, ξi does notconverge a.s. to ξ.

Section 1.14 - Conditional Probability 39

Convergence in Probability vs. Convergence in Distribution

Theorem 1.44 Suppose that ξ, ξ1, ξ2, · · · are random variables defined onthe probability space (Ω,A,Pr). If the sequence ξi converges in probabilityto ξ, then ξi converges in distribution to ξ.

Proof: Let x be any given continuity point of the distribution Φ. On theone hand, for any y > x, we have

ξi ≤ x = ξi ≤ x, ξ ≤ y ∪ ξi ≤ x, ξ > y ⊂ ξ ≤ y ∪ |ξi − ξ| ≥ y − x

which implies that

Φi(x) ≤ Φ(y) + Pr|ξi − ξ| ≥ y − x.

Since ξi converges in probability to ξ, we have Pr|ξi − ξ| ≥ y − x → 0.Thus we obtain lim supi→∞ Φi(x) ≤ Φ(y) for any y > x. Letting y → x, weget

lim supi→∞

Φi(x) ≤ Φ(x). (1.59)

On the other hand, for any z < x, we have

ξ ≤ z = ξ ≤ z, ξi ≤ x ∪ ξ ≤ z, ξi > x ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) + Pr|ξi − ξ| ≥ x− z.

Since Pr|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞ Φi(x) for anyz < x. Letting z → x, we get

Φ(x) ≤ lim infi→∞

Φi(x). (1.60)

It follows from (1.59) and (1.60) that Φi(x) → Φ(x). The theorem is proved.

Example 1.31: Convergence in distribution does not imply convergencein probability. For example, take (Ω,A,Pr) to be ω1, ω2 with Prω1 =Prω2 = 0.5, and

ξ(ω) =−1, if ω = ω1

1, if ω = ω2.

We also define ξi = −ξ for all i. Then ξi and ξ are identically distributed.Thus ξi converges in distribution to ξ. But, for any small number ε > 0,we have Pr|ξi − ξ| > ε = PrΩ = 1. That is, the sequence ξi does notconverge in probability to ξ.


1.14 Conditional Probability

We consider the probability of an event A after it has been learned thatsome other event B has occurred. This new probability of A is called theconditional probability of A given B.

Definition 1.28 Let (Ω,A,Pr) be a probability space, and A,B ∈ A. Thenthe conditional probability of A given B is defined by

PrA|B =PrA ∩B

PrB(1.61)

provided that PrB > 0.

Theorem 1.45 Let (Ω,A,Pr) be a probability space, and B an event withPrB > 0. Then Pr·|B defined by (1.61) is a probability measure, and(Ω,A,Pr·|B) is a probability space.

Proof: It is sufficient to prove that Pr·|B satisfies the normality, nonneg-ativity and countable additivity axioms. At first, we have

PrΩ|B =PrΩ ∩B

PrB=

PrBPrB

= 1.

Secondly, for any A ∈ A, the set function PrA|B is nonnegative. Finally,for any countable sequence Ai of mutually disjoint events, we have

Pr

∞⋃i=1

Ai|B

=

Pr( ∞⋃

i=1

Ai

)∩B

PrB

=

∞∑i=1

PrAi ∩B

PrB=

∞∑i=1

PrAi|B.

Thus Pr·|B is a probability measure. Furthermore, (Ω,A,Pr·|B) is aprobability space.

Theorem 1.46 (Bayes Formula) Let the events A1, A2, · · · , An form a par-tition of the space Ω such that PrAi > 0 for i = 1, 2, · · · , n, and let B bean event with PrB > 0. Then we have

PrAk|B =PrAkPrB|Akn∑

i=1

PrAiPrB|Ai(1.62)

for k = 1, 2, · · · , n.

Proof: Since A1, A2, · · · , An form a partition of the space Ω, we have

PrB =n∑

i=1

PrAi ∩B =n∑

i=1

PrAiPrB|Ai

Section 1.14 - Conditional Probability 41

which is also called the formula for total probability. Thus, for any k, we have

PrAk|B =PrAk ∩B

PrB=

PrAkPrB|Akn∑

i=1

PrAiPrB|Ai.


Remark 1.2: Especially, let A and B be two events with PrA > 0 andPrB > 0. Then A and Ac form a partition of the space Ω, and the Bayesformula is

PrA|B =PrAPrB|A

PrB. (1.63)

Remark 1.3: In statistical applications, the events A1, A2, · · · , An are oftencalled hypotheses. Furthermore, for each i, the PrAi is called the priorprobability of Ai, and PrAi|B is called the posterior probability of Ai afterthe occurrence of event B.

Example 1.32: Let ξ be an exponentially distributed random variable withexpected value β. Then for any real numbers a > 0 and x > 0, the conditionalprobability of ξ ≥ a+ x given ξ ≥ a is

Prξ ≥ a+ x|ξ ≥ a = exp(−x/β) = Prξ ≥ x

which means that the conditional probability is identical to the original prob-ability. This is the so-called memoryless property of exponential distribution.In other words, it is as good as new if it is functioning on inspection.

Definition 1.29 The conditional probability distribution Φ: < → [0, 1] of arandom variable ξ given B is defined by

Φ(x|B) = Pr ξ ≤ x|B (1.64)

provided that PrB > 0.

Example 1.33: Let ξ and η be random variables. Then the conditionalprobability distribution of ξ given η = y is

Φ(x|η = y) = Pr ξ ≤ x|η = y =Prξ ≤ x, η = y

Prη = y

provided that Prη = y > 0.

Definition 1.30 The conditional probability density function φ of a randomvariable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ < (1.65)

where Φ(x|B) is the conditional probability distribution of ξ given B.


Example 1.34: Let (ξ, η) be a random vector with joint probability densityfunction ψ. Then the marginal probability density functions of ξ and η are

f(x) =∫ +∞

−∞ψ(x, y)dy, g(y) =

∫ +∞

−∞ψ(x, y)dx,

respectively. Furthermore, we have

Prξ ≤ x, η ≤ y =∫ x

−∞

∫ y

−∞ψ(r, t)drdt =

∫ y

−∞

[∫ x

−∞

ψ(r, t)g(t)

dr]g(t)dt

which implies that the conditional probability distribution of ξ given η = yis

Φ(x|η = y) =∫ x

−∞

ψ(r, y)g(y)

dr, a.s. (1.66)

and the conditional probability density function of ξ given η = y is

φ(x|η = y) =ψ(x, y)g(y)

=ψ(x, y)∫ +∞

−∞ψ(x, y)dx

, a.s. (1.67)

Note that (1.66) and (1.67) are defined only for g(y) 6= 0. In fact, the sety|g(y) = 0 has probability 0. Especially, if ξ and η are independent randomvariables, then ψ(x, y) = f(x)g(y) and φ(x|η = y) = f(x).

Definition 1.31 Let ξ be a random variable. Then the conditional expectedvalue of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Prξ ≥ r|Bdr −∫ 0

−∞Prξ ≤ r|Bdr (1.68)


Following conditional probability and conditional expected value, we alsohave conditional variance, conditional moments, conditional critical values,conditional entropy as well as conditional convergence.

Definition 1.32 Let ξ be a nonnegative random variable representing life-time. Then the hazard rate (or failure rate) is

h(x) = lim∆↓0

Prξ ≤ x+ ∆∣∣ ξ > x

∆. (1.69)

The hazard rate tells us the probability of a failure just after time x whenit is functioning at time x. If ξ has probability distribution Φ and probabilitydensity function φ, then the hazard rate

h(x) =φ(x)

1− Φ(x).

Example 1.35: Let ξ be an exponentially distributed random variable withexpected value β. Then its hazard rate h(x) ≡ 1/β.

Section 1.15 - Stochastic Process 43

1.15 Stochastic Process

Definition 1.33 Let T be an index set and (Ω,A,Pr) be a probability space.A stochastic process is a measurable function from T × (Ω,A,Pr) to the setof real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, theset

ω ∈ Ω∣∣ X(t, ω) ∈ B (1.70)

is an event.

That is, a stochastic process Xt(ω) is a function of two variables suchthat the function Xt∗(ω) is a random variable for each t∗. For each fixed ω∗,the function Xt(ω∗) is said to be a sample path of the stochastic process. Astochastic process Xt(ω) is said to be sample-continuous if the sample pathis continuous for almost all ω.

Definition 1.34 A stochastic process Xt is said to have independent incre-ments if

Xt1 −Xt0 , Xt2 −Xt1 , · · · , Xtk−Xtk−1 (1.71)

are independent random variables for any times t0 < t1 < · · · < tk. Astochastic process Xt is said to have stationary increments if, for any givent > 0, the increments Xs+t −Xs are identically distributed random variablesfor all s > 0.

Renewal Process

Definition 1.35 Let ξ1, ξ2, · · · be iid positive random variables. Define S0 =0 and Sn = ξ1 + ξ2 + · · ·+ ξn for n ≥ 1. Then the stochastic process

Nt = maxn≥0

n∣∣ Sn ≤ t

(1.72)

is called a renewal process.

If ξ1, ξ2, · · · denote the interarrival times of successive events. Then Sn

can be regarded as the waiting time until the occurrence of the nth event,and Nt is the number of renewals in (0, t]. Each sample path of Nt is aright-continuous and increasing step function taking only nonnegative integervalues. Furthermore, the size of each jump of Nt is always 1. In other words,Nt has at most one renewal at each time. In particular, Nt does not jump attime 0. Since Nt ≥ n if and only if Sn ≤ t, we have

PrNt ≥ n = PrSn ≤ t. (1.73)

Theorem 1.47 Let Nt be a renewal process. Then we have

E[Nt] =∞∑

n=1

PrSn ≤ t. (1.74)


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...............

4 ...............

3 ...............

2 ...............

1 ...............

0 ...............

ξ1 ξ2 ξ3 ξ4

S0 S1 S2 S3 S4

...........................................

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t

Nt

...........................................................................................................

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...........................

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.......................................................................................................

Figure 1.2: A Sample Path of Renewal Process

Proof: Since Nt takes only nonnegative integer values, we have

E[Nt] =∫ ∞

0

PrNt ≥ rdr =∞∑

n=1

∫ n

n−1

PrNt ≥ rdr

=∞∑

n=1

PrNt ≥ n =∞∑

n=1

PrSn ≤ t.


Example 1.36: A renewal process Nt is called a Poisson process with in-tensity λ if ξ1, ξ2, · · · are iid exponentially distributed random variables withexpected value 1/λ. It has been proved that

PrNt = n = exp(−λt) (λt)n

n!, n = 0, 1, 2, · · · (1.75)

E[Nt] = λt. (1.76)

Theorem 1.48 (Renewal Theorem) Let Nt be a renewal process with inter-arrival times ξ1, ξ2, · · · Then we have

limt→∞

E[Nt]t

=1

E[ξ1]. (1.77)

Brownian Motion

In 1828 the botanist Brown observed irregular movement of pollen suspendedin liquid. This movement is now known as Brownian motion. Bachelier usedBrownian motion as a model of stock prices in 1900. An equation for Brow-nian motion was obtained by Einstein in 1905, and a rigorous mathematicaldefinition of Brownian motion was given by Wiener in 1931. For this reason,Brownian motion is also called Wiener process.

Section 1.15 - Stochastic Process 45

Definition 1.36 A stochastic process Bt is said to be a Brownian motion(also called Wiener process) if(i) B0 = 0 and Bt is sample-continuous,(ii) Bt has stationary and independent increments,(iii) every increment Bs+t − Bs is a normally distributed random variablewith expected value et and variance σ2t.

The parameters e and σ are called the drift and diffusion coefficients,respectively. The Brownian motion is said to be standard if e = 0 and σ = 1.Any Brownian motion may be represented by et+σBt where Bt is a standardBrownian motion.

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t

Bt

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Figure 1.3: A Sample Path of Standard Brownian Motion

Theorem 1.49 (Existence Theorem) There is a Brownian motion.

Proof: Without loss of generality, we only prove that there is a standardBrownian motion Bt on the range of t ∈ [0, 1]. First, let

ξ(r)∣∣ r represents rational numbers in [0, 1]

be a countable sequence of independently and normally distributed randomvariables with expected value zero and variance one. For each integer n, wedefine a stochastic process

Xn(t) =

1√n

k∑i=1

ξ

(i

n

), if t =

k

n(k = 0, 1, · · · , n)

linear, otherwise.

Since the limitlim

n→∞Xn(t)

exists almost surely, we may verify that the limit meets the conditions ofstandard Brownian motion. Hence there is a standard Brownian motion.


Remark 1.4: Suppose that Bt is a standard Brownian motion. It has beenproved that

X1(t) = −Bt, (1.78)

X2(t) = aBt/a2 , (1.79)

X3(t) = Bt+s −Bs (1.80)

are each a version of standard Brownian motion.Almost all Brownian paths have an infinite variation and are differentiable

nowhere. Furthermore, the squared variation of Brownian motion on [0, t] isequal to t both in mean square and almost surely.

On the one hand, almost all Brownian paths are infinitely long within anytime interval. On the other hand, it is impossible that a pollen moves at aspeed of infinity. Is it a dilemma?

Example 1.37: Let Bt be a Brownian motion with drift 0. Then for anylevel x > 0 and any time t > 0, we have

Pr

max0≤s≤t

Bs ≥ x

= 2PrBt ≥ x (1.81)

which is the so-called reflection principle. For any level x < 0 and any timet > 0, we have

Pr

min0≤s≤t

Bs ≤ x

= 2PrBt ≤ x. (1.82)

Example 1.38: Let Bt be a Brownian motion with drift e > 0 and diffusioncoefficient σ. Then the first passage time τ that the Brownian motion reachesthe barrier x > 0 has the probability density function

φ(t) =x

σ√

2πt3exp

(− (x− et)2

2σ2t

), t > 0 (1.83)

whose expected value and variance are

E[τ ] =x

e, V [τ ] =

xσ2

e3. (1.84)

However, if the drift e = 0, then the expected value E[τ ] is infinite.

Definition 1.37 Let Bt be a standard Brownian motion. Then et + σBt isa Brownian motion, and the stochastic process

Gt = exp(et+ σBt) (1.85)

is called a geometric Brownian motion.

Section 1.16 - Stochastic Calculus 47

Geometric Brownian motion Gt is an important model for stock prices.For each t > 0, the Gt has a lognormal distribution whose probability densityfunction is

φ(z) =1

σ√

2πtexp

(− (ln z − et)2

2σ2t

), z ≥ 0 (1.86)

and has expected value and variance as follows,

E[Gt] = exp(et+ σ2t/2

),

V [Gt] = exp(2et+ 2σ2t)− exp(2et+ σ2t).

In addition, the first passage time that a geometric Brownian motion Gt

reaches the barrier x > 1 is just the time that the Brownian motion withdrift e and diffusion σ reaches lnx.

1.16 Stochastic Calculus

Let Bt be a standard Brownian motion, and dt an infinitesimal time interval.Then

dBt = Bt+dt −Bt

is a stochastic process such that, for each t, the dBt is a normally distributedrandom variable with

E[dBt] = 0, V [dBt] = dt,

E[dB2t ] = dt, V [dB2

t ] = 2dt2.

Definition 1.38 Let Xt be a stochastic process and let Bt be a standardBrownian motion. For any partition of closed interval [a, b] with a = t1 <t2 < · · · < tk+1 = b, the mesh is written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the Ito integral of Xt with respect to Bt is∫ b

a

XtdBt = lim∆→0

k∑i=1

Xti · (Bti+1 −Bti) (1.87)

provided that the limit exists in mean square and is a random variable.

Example 1.39: Let Bt be a standard Brownian motion. Then for anypartition 0 = t1 < t2 < · · · < tk+1 = s, we have∫ s

0

dBt = lim∆→0

k∑i=1

(Bti+1 −Bti) ≡ Bs −B0 = Bs.


Example 1.40: Let Bt be a standard Brownian motion. Then for anypartition 0 = t1 < t2 < · · · < tk+1 = s, we have

sBs =k∑

i=1

(ti+1Bti+1 − tiBti

)=

k∑i=1

ti(Bti+1 −Bti) +

k∑i=1

Bti+1(ti+1 − ti)

→∫ s

0

tdBt +∫ s

0

Btdt

as ∆ → 0. It follows that∫ s

0

tdBt = sBs −∫ s

0

Btdt.

Example 1.41: Let Bt be a standard Brownian motion. Then for anypartition 0 = t1 < t2 < · · · < tk+1 = s, we have

B2s =

k∑i=1

(B2

ti+1−B2

ti

)

=k∑

i=1

(Bti+1 −Bti

)2 + 2k∑

i=1

Bti

(Bti+1 −Bti

)→ s+ 2

∫ s

0

BtdBt

as ∆ → 0. That is, ∫ s

0

BtdBt =12B2

s −12s.

Theorem 1.50 (Ito Formula) Let Bt be a standard Brownian motion, andlet h(t, b) be a twice continuously differentiable function. Define Xt = h(t, Bt).Then we have the following chain rule

dXt =∂h

∂t(t, Bt)dt+

∂h

∂b(t, Bt)dBt +

12∂2h

∂b2(t, Bt)dt. (1.88)

Proof: Since the function h is twice continuously differentiable, by usingTaylor series expansion, the infinitesimal increment of Xt has a second-orderapproximation

∆Xt =∂h

∂t(t, Bt)∆t+

∂h

∂b(t, Bt)∆Bt +

12∂2h

∂b2(t, Bt)(∆Bt)2

+12∂2h

∂t2(t, Bt)(∆t)2 +

∂2h

∂t∂b(t, Bt)∆t∆Bt.

Section 1.16 - Stochastic Calculus 49

Since we can ignore the terms (∆t)2 and ∆t∆Bt and replace (∆Bt)2 with∆t, the Ito formula is obtained because it makes

Xs = X0 +∫ s

0

∂h

∂t(t, Bt)dt+

∫ s

0

∂h

∂b(t, Bt)dBt +

12

∫ s

0

∂2h

∂b2(t, Bt)dt

for any s ≥ 0.

Remark 1.5: The infinitesimal increment dBt in (1.88) may be replacedwith the Ito process

dYt = utdt+ vtdBt (1.89)

where ut is an absolutely integrable stochastic process, and vt is a squareintegrable stochastic process, thus producing

dh(t, Yt) =∂h

∂t(t, Yt)dt+

∂h

∂b(t, Yt)dYt +

12∂2h

∂b2(t, Yt)v2

t dt. (1.90)

Remark 1.6: Assume that B1t, B2t, · · · , Bmt are standard Brownian mo-tions, and h(t, b1, b2, · · · , bm) is a twice continuously differentiable function.Define

Xt = h(t, B1t, B2t, · · · , Bmt).

Then we have the following multi-dimensional Ito formula

dXt =∂h

∂tdt+

m∑i=1

∂h

∂bidBit +

12

m∑i=1

∂2h

∂b2idt. (1.91)

Example 1.42: Ito formula is the chain rule for differentiation. ApplyingIto formula, we obtain

d(tBt) = Btdt+ tdBt.

Hence we have

sBs =∫ s

0

d(tBt) =∫ s

0

Btdt+∫ s

0

tdBt.

That is, ∫ s

0

tdBt = sBs −∫ s

0

Btdt.

Example 1.43: Let Bt be a standard Brownian motion. By using Itoformula

d(B2t ) = 2BtdBt + dt,

we obtain

B2s =

∫ s

0

d(B2t ) = 2

∫ s

0

BtdBt +∫ s

0

dt = 2∫ s

0

BtdBt + s.


It follows that ∫ s

0

BtdBt =12B2

s −12s.

Example 1.44: Let Bt be a standard Brownian motion. By using Itoformula

d(B3t ) = 3B2

t dBt + 3Btdt,

we haveB3

s =∫ s

0

d(B3t ) = 3

∫ s

0

B2t dBt + 3

∫ s

0

Btdt.

That is ∫ s

0

B2t dBt =

13B3

s −∫ s

0

Btdt.

Theorem 1.51 (Integration by Parts) Suppose that Bt is a standard Brow-nian motion and F (t) is an absolutely continuous function. Then∫ s

0

F (t)dBt = F (s)Bs −∫ s

0

BtdF (t). (1.92)

Proof: By defining h(t, Bt) = F (t)Bt and using the Ito formula, we get

d(F (t)Bt) = BtdF (t) + F (t)dBt.

ThusF (s)Bs =

∫ s

0

d(F (t)Bt) =∫ s

0

BtdF (t) +∫ s

0

F (t)dBt

which is just (1.92).

1.17 Stochastic Differential Equation

This section introduces a type of stochastic differential equations driven byBrownian motion.

Definition 1.39 Suppose Bt is a standard Brownian motion, and f and gare some given functions. Then

dXt = f(t,Xt)dt+ g(t,Xt)dBt (1.93)

is called a stochastic differential equation. A solution is a stochastic processXt that satisfies (1.93) identically in t.

Remark 1.7: Note that there is no precise definition for the terms dXt, dtand dBt in the stochastic differential equation (1.93). The mathematicallymeaningful form is the stochastic integral equation

Xs = X0 +∫ s

0

f(t,Xt)dt+∫ s

0

g(t,Xt)dBt. (1.94)

Section 1.17 - Stochastic Differential Equation 51

However, the differential form is convenient for us. This is the main reasonwhy we accept the differential form.

Example 1.45: Let Bt be a standard Brownian motion. Then the stochasticdifferential equation

dXt = adt+ bdBt

has a solutionXt = at+ bBt

which is just a Brownian motion with drift coefficient a and diffusion coeffi-cient b.

Example 1.46: Let Bt be a standard Brownian motion. Then the stochasticdifferential equation

dXt = aXtdt+ bXtdBt

has a solution

Xt = exp((

a− b2

2

)t+ bBt

)which is just a geometric Brownian motion.

Example 1.47: Let Bt be a standard Brownian motion. Then the stochasticdifferential equations

dXt = −12Xtdt− YtdBt

dYt = −12Ytdt+XtdBt

have a solution(Xt, Yt) = (cosBt, sinBt)

which is called a Brownian motion on unit circle since X2t + Y 2

t ≡ 1.

Chapter 2

Credibility Theory

The concept of fuzzy set was initiated by Zadeh [245] via membership functionin 1965. In order to measure a fuzzy event, Zadeh [248] proposed the conceptof possibility measure. Although possibility measure has been widely used,it has no self-duality property. However, a self-dual measure is absolutelyneeded in both theory and practice. In order to define a self-dual measure,Liu and Liu [126] presented the concept of credibility measure. In addition,a sufficient and necessary condition for credibility measure was given by Liand Liu [100].

Credibility theory, founded by Liu [129] in 2004 and refined by Liu [132] in2007, is a branch of mathematics for studying the behavior of fuzzy phenom-ena. The emphasis in this chapter is mainly on credibility measure, credibil-ity space, fuzzy variable, membership function, credibility distribution, inde-pendence, identical distribution, expected value, variance, moments, criticalvalues, entropy, distance, convergence almost surely, convergence in credibil-ity, convergence in mean, convergence in distribution, conditional credibility,fuzzy process, fuzzy calculus, and fuzzy differential equation.

2.1 Credibility Space

Let Θ be a nonempty set, and P the power set of Θ (i.e., the larggest σ-algebra over Θ). Each element in P is called an event. In order to present anaxiomatic definition of credibility, it is necessary to assign to each event A anumber CrA which indicates the credibility that A will occur. In order toensure that the number CrA has certain mathematical properties which weintuitively expect a credibility to have, we accept the following four axioms:

Axiom 1. (Normality) CrΘ = 1.

Axiom 2. (Monotonicity) CrA ≤ CrB whenever A ⊂ B.

Axiom 3. (Self-Duality) CrA+ CrAc = 1 for any event A.

54 Chapter 2 - Credibility Theory

Axiom 4. (Maximality) Cr ∪iAi = supi CrAi for any events Ai withsupi CrAi < 0.5.

Definition 2.1 (Liu and Liu [126]) The set function Cr is called a cred-ibility measure if it satisfies the normality, monotonicity, self-duality, andmaximality axioms.

Example 2.1: Let Θ = θ1, θ2. For this case, there are only four events:∅, θ1, θ2,Θ. Define Cr∅ = 0, Crθ1 = 0.7, Crθ2 = 0.3, and CrΘ =1. Then the set function Cr is a credibility measure because it satisfies thefour axioms.

Example 2.2: Let Θ be a nonempty set. Define Cr∅ = 0, CrΘ = 1 andCrA = 1/2 for any subset A (excluding ∅ and Θ). Then the set functionCr is a credibility measure.

Theorem 2.1 Let Θ be a nonempty set, P the power set of Θ, and Cr thecredibility measure. Then Cr∅ = 0 and 0 ≤ CrA ≤ 1 for any A ∈ P.

Proof: It follows from Axioms 1 and 3 that Cr∅ = 1−CrΘ = 1−1 = 0.Since ∅ ⊂ A ⊂ Θ, we have 0 ≤ CrA ≤ 1 by using Axiom 2.

Theorem 2.2 Let Θ be a nonempty set, P the power set of Θ, and Cr thecredibility measure. Then for any A,B ∈ P, we have

CrA ∪B = CrA ∨ CrB if CrA ∪B ≤ 0.5, (2.1)

CrA ∩B = CrA ∧ CrB if CrA ∩B ≥ 0.5. (2.2)

The above equations hold for not only finite number of events but also infinitenumber of events.

Proof: If CrA ∪ B < 0.5, then CrA ∨ CrB < 0.5 by using Axiom 2.Thus the equation (2.1) follows immediately from Axiom 4. If CrA ∪B =0.5 and (2.1) does not hold, then we have CrA ∨ CrB < 0.5. It followsfrom Axiom 4 that

CrA ∪B = CrA ∨ CrB < 0.5.

A contradiction proves (2.1). Next we prove (2.2). Since CrA ∩ B ≥ 0.5,we have CrAc ∪Bc ≤ 0.5 by the self-duality. Thus

CrA ∩B = 1− CrAc ∪Bc = 1− CrAc ∨ CrBc = CrA ∧ CrB.


Section 2.1 - Credibility Space 55

Credibility Subadditivity Theorem

Theorem 2.3 (Liu [129], Credibility Subadditivity Theorem) The credibilitymeasure is subadditive. That is,

CrA ∪B ≤ CrA+ CrB (2.3)

for any events A and B. In fact, credibility measure is not only finitelysubadditive but also countably subadditive.

Proof: The argument breaks down into three cases.Case 1: CrA < 0.5 and CrB < 0.5. It follows from Axiom 4 that

CrA ∪B = CrA ∨ CrB ≤ CrA+ CrB.

Case 2: CrA ≥ 0.5. For this case, by using Axioms 2 and 3, we haveCrAc ≤ 0.5 and CrA ∪B ≥ CrA ≥ 0.5. Then

CrAc = CrAc ∩B ∨ CrAc ∩Bc

≤ CrAc ∩B+ CrAc ∩Bc

≤ CrB+ CrAc ∩Bc.

Applying this inequality, we obtain

CrA+ CrB = 1− CrAc+ CrB

≥ 1− CrB − CrAc ∩Bc+ CrB

= 1− CrAc ∩Bc

= CrA ∪B.

Case 3: CrB ≥ 0.5. This case may be proved by a similar process ofCase 2. The theorem is proved.

Remark 2.1: For any eventsA andB, it follows from the credibility subaddi-tivity theorem that the credibility measure is null-additive, i.e., CrA∪B =CrA+ CrB if either CrA = 0 or CrB = 0.

Theorem 2.4 Let Bi be a decreasing sequence of events with CrBi → 0as i→∞. Then for any event A, we have

limi→∞

CrA ∪Bi = limi→∞

CrA\Bi = CrA. (2.4)

Proof: It follows from the monotonicity axiom and credibility subadditivitytheorem that

CrA ≤ CrA ∪Bi ≤ CrA+ CrBi


for each i. Thus we get CrA ∪ Bi → CrA by using CrBi → 0. Since(A\Bi) ⊂ A ⊂ ((A\Bi) ∪Bi), we have

CrA\Bi ≤ CrA ≤ CrA\Bi+ CrBi.

Hence CrA\Bi → CrA by using CrBi → 0.

Theorem 2.5 A credibility measure on Θ is additive if and only if there areat most two elements in Θ taking nonzero credibility values.

Proof: Suppose that the credibility measure is additive. If there are morethan two elements taking nonzero credibility values, then we may choose threeelements θ1, θ2, θ3 such that Crθ1 ≥ Crθ2 ≥ Crθ3 > 0. If Crθ1 ≥ 0.5,it follows from Axioms 2 and 3 that

Crθ2, θ3 ≤ CrΘ \ θ1 = 1− Crθ1 ≤ 0.5.

By using Axiom 4, we obtain

Crθ2, θ3 = Crθ2 ∨ Crθ3 < Crθ2+ Crθ3.

This is in contradiction with the additivity assumption. If Crθ1 < 0.5,then Crθ3 ≤ Crθ2 < 0.5. It follows from Axiom 4 that

Crθ2, θ3 ∧ 0.5 = Crθ2 ∨ Crθ3 < 0.5

which implies that

Crθ2, θ3 = Crθ2 ∨ Crθ3 < Crθ2+ Crθ3.

This is also in contradiction with the additivity assumption. Hence there areat most two elements taking nonzero credibility values.

Conversely, suppose that there are at most two elements, say θ1 and θ2,taking nonzero credibility values. Let A and B be two disjoint events. Theargument breaks down into two cases.

Case 1: If either CrA = 0 or CrB = 0 is true, then we have CrA ∪B = CrA+ CrB by using the credibility subadditivity theorem.

Case 2: CrA > 0 or CrB > 0. For this case, without loss of generality,we suppose that θ1 ∈ A and θ2 ∈ B. Note that Cr(A ∪B)c = 0. It followsfrom Axiom 3 and the credibility subadditivity theorem that

CrA ∪B = CrA ∪B ∪ (A ∪B)c = CrΘ = 1,

CrA+ CrB = CrA ∪ (A ∪B)c+ CrB = 1.

Hence CrA ∪B = CrA+ CrB. The additivity is proved.

Remark 2.2: Theorem 2.5 states that a credibility measure is identical withprobability measure if there are effectively two elements in the universal set.


Credibility Semicontinuity Law

Generally speaking, the credibility measure is neither lower semicontinuousnor upper semicontinuous. However, we have the following credibility semi-continuity law.

Theorem 2.6 (Liu [129], Credibility Semicontinuity Law) For any eventsA1, A2, · · · , we have

limi→∞

CrAi = Cr

limi→∞

Ai

(2.5)

if one of the following conditions is satisfied:(a) Cr A ≤ 0.5 and Ai ↑ A; (b) lim

i→∞CrAi < 0.5 and Ai ↑ A;

(c) Cr A ≥ 0.5 and Ai ↓ A; (d) limi→∞

CrAi > 0.5 and Ai ↓ A.

Proof: (a) Since CrA ≤ 0.5, we have CrAi ≤ 0.5 for each i. It followsfrom Axiom 4 that

CrA = Cr ∪iAi = supi

CrAi = limi→∞

CrAi.

(b) Since limi→∞ CrAi < 0.5, we have supi CrAi < 0.5. It followsfrom Axiom 4 that

CrA = Cr ∪iAi = supi

CrAi = limi→∞

CrAi.

(c) Since CrA ≥ 0.5 and Ai ↓ A, it follows from the self-duality ofcredibility measure that CrAc ≤ 0.5 and Ac

i ↑ Ac. Thus CrAi = 1 −CrAc

i → 1− CrAc = CrA as i→∞.(d) Since limi→∞ CrAi > 0.5 and Ai ↓ A, it follows from the self-duality

of credibility measure that

limi→∞

CrAci = lim

i→∞(1− CrAi) < 0.5

and Aci ↑ Ac. Thus CrAi = 1−CrAc

i → 1−CrAc = CrA as i→∞.The theorem is proved.

Credibility Asymptotic Theorem

Theorem 2.7 (Credibility Asymptotic Theorem) For any events A1, A2, · · · ,we have

limi→∞

CrAi ≥ 0.5, if Ai ↑ Θ, (2.6)

limi→∞

CrAi ≤ 0.5, if Ai ↓ ∅. (2.7)


Proof: Assume Ai ↑ Θ. If limi→∞ CrAi < 0.5, it follows from the credi-bility semicontinuity law that

CrΘ = limi→∞

CrAi < 0.5

which is in contradiction with CrΘ = 1. The first inequality is proved.The second one may be verified similarly.

Credibility Extension Theorem

Suppose that the credibility of each singleton is given. Is the credibilitymeasure fully and uniquely determined? This subsection will answer thequestion.

Theorem 2.8 Suppose that Θ is a nonempty set. If Cr is a credibility mea-sure, then we have

supθ∈Θ

Crθ ≥ 0.5,

Crθ∗+ supθ 6=θ∗

Crθ = 1 if Crθ∗ ≥ 0.5.(2.8)

We will call (2.8) the credibility extension condition.

Proof: If supCrθ < 0.5, then by using Axiom 4, we have

1 = CrΘ = supθ∈Θ

Crθ < 0.5.

This contradiction proves supCrθ ≥ 0.5. We suppose that θ∗ ∈ Θ is a pointwith Crθ∗ ≥ 0.5. It follows from Axioms 3 and 4 that CrΘ \ θ∗ ≤ 0.5,and

CrΘ \ θ∗ = supθ 6=θ∗

Crθ.

Hence the second formula of (2.8) is true by the self-duality of credibilitymeasure.

Theorem 2.9 (Li and Liu [100], Credibility Extension Theorem) Supposethat Θ is a nonempty set, and Crθ is a nonnegative function on Θ satisfyingthe credibility extension condition (2.8). Then Crθ has a unique extensionto a credibility measure as follows,

CrA =

supθ∈A

Crθ, if supθ∈A

Crθ < 0.5

1− supθ∈Ac

Crθ, if supθ∈A

Crθ ≥ 0.5.(2.9)


Proof: We first prove that the set function CrA defined by (2.9) is acredibility measure.

Step 1: By the credibility extension condition supθ∈Θ

Crθ ≥ 0.5, we have

CrΘ = 1− supθ∈∅

Crθ = 1− 0 = 1.

Step 2: If A ⊂ B, then Bc ⊂ Ac. The proof breaks down into two cases.Case 1: sup

θ∈ACrθ < 0.5. For this case, we have

CrA = supθ∈A

Crθ ≤ supθ∈B

Crθ ≤ CrB.

Case 2: supθ∈A

Crθ ≥ 0.5. For this case, we have supθ∈B

Crθ ≥ 0.5, and

CrA = 1− supθ∈Ac

Crθ ≤ 1− supθ∈Bc

Crθ = CrB.

Step 3: In order to prove CrA + CrAc = 1, the argument breaksdown into two cases.

Case 1: supθ∈A

Crθ < 0.5. For this case, we have supθ∈Ac

Crθ ≥ 0.5. Thus,

CrA+ CrAc = supθ∈A

Crθ+ 1− supθ∈A

Crθ = 1.

Case 2: supθ∈A

Crθ ≥ 0.5. For this case, we have supθ∈Ac

Crθ ≤ 0.5, and

CrA+ CrAc = 1− supθ∈Ac

Crθ+ supθ∈Ac

Crθ = 1.

Step 4: For any collection Ai with supi CrAi < 0.5, we have

Cr∪iAi = supθ∈∪iAi

Crθ = supi

supθ∈Ai

Crθ = supi

CrAi.

Thus Cr is a credibility measure because it satisfies the four axioms.Finally, let us prove the uniqueness. Assume that Cr1 and Cr2 are two

credibility measures such that Cr1θ = Cr2θ for each θ ∈ Θ. Let us provethat Cr1A = Cr2A for any event A. The argument breaks down intothree cases.

Case 1: Cr1A < 0.5. For this case, it follows from Axiom 4 that

Cr1A = supθ∈A

Cr1θ = supθ∈A

Cr2θ = Cr2A.

Case 2: Cr1A > 0.5. For this case, we have Cr1Ac < 0.5. It followsfrom the first case that Cr1Ac = Cr2Ac which implies Cr1A = Cr2A.

Case 3: Cr1A = 0.5. For this case, we have Cr1Ac = 0.5, and

Cr2A ≥ supθ∈A

Cr2θ = supθ∈A

Cr1θ = Cr1A = 0.5,

Cr2Ac ≥ supθ∈Ac

Cr2θ = supθ∈Ac

Cr1θ = Cr1Ac = 0.5.

Hence Cr2A = 0.5 = Cr1A. The uniqueness is proved.


Credibility Space

Definition 2.2 Let Θ be a nonempty set, P the power set of Θ, and Cr acredibility measure. Then the triplet (Θ,P,Cr) is called a credibility space.

Example 2.3: The triplet (Θ,P,Cr) is a credibility space if

Θ = θ1, θ2, · · · , Crθi ≡ 1/2 for i = 1, 2, · · · (2.10)

Note that the credibility measure is produced by the credibility extensiontheorem as follows,

CrA =

0, if A = ∅1, if A = Θ

1/2, otherwise.


Θ = θ1, θ2, · · · , Crθi = i/(2i+ 1) for i = 1, 2, · · · (2.11)

By using the credibility extension theorem, we obtain the following credibilitymeasure,

CrA =

supθi∈A

i

2i+ 1, if A is finite

1− supθi∈Ac

i

2i+ 1, if A is infinite.


Θ = θ1, θ2, · · · , Crθ1 = 1/2, Crθi = 1/i for i = 2, 3, · · · (2.12)

For this case, the credibility measure is

CrA =

supθi∈A

1/i, if A contains neither θ1 nor θ2

1/2, if A contains only one of θ1 and θ2

1− supθi∈Ac

1/i, if A contains both θ1 and θ2.


Θ = [0, 1], Crθ = θ/2 for θ ∈ Θ. (2.13)

For this case, the credibility measure is

CrA =

12

supθ∈A

θ, if supθ∈A

θ < 1

1− 12

supθ∈Ac

θ, if supθ∈A

θ = 1.


Product Credibility Measure

Product credibility measure may be defined in multiple ways. This bookaccepts the following axiom.

Axiom 5. (Product Credibility Axiom) Let Θk be nonempty sets on whichCrk are credibility measures, k = 1, 2, · · · , n, respectively, and Θ = Θ1×Θ2×· · · ×Θn. Then

Cr(θ1, θ2, · · · , θn) = Cr1θ1 ∧ Cr2θ2 ∧ · · · ∧ Crnθn (2.14)

for each (θ1, θ2, · · · , θn) ∈ Θ.

Theorem 2.10 (Product Credibility Theorem) Let Θk be nonempty sets onwhich Crk are the credibility measures, k = 1, 2, · · · , n, respectively, and Θ =Θ1×Θ2×· · ·×Θn. Then Cr = Cr1 ∧Cr2 ∧ · · · ∧Crn defined by Axiom 5 hasa unique extension to a credibility measure on Θ as follows,

CrA =

sup(θ1,θ2··· ,θn)∈A

min1≤k≤n

Crkθk,

if sup(θ1,θ2,··· ,θn)∈A

min1≤k≤n

Crkθk < 0.5

1− sup(θ1,θ2,··· ,θn)∈Ac

min1≤k≤n

Crkθk,

if sup(θ1,θ2,··· ,θn)∈A

min1≤k≤n

Crkθk ≥ 0.5.

(2.15)

Proof: For each θ = (θ1, θ2, · · · , θn) ∈ Θ, we have Crθ = Cr1θ1 ∧Cr2θ2 ∧ · · · ∧ Crnθn. Let us prove that Crθ satisfies the credibilityextension condition. Since sup

θk∈Θk

Crθk ≥ 0.5 for each k, we have

supθ∈Θ

Crθ = sup(θ1,θ2,··· ,θn)∈Θ

min1≤k≤n

Crkθk ≥ 0.5.

Now we suppose that θ∗ = (θ∗1 , θ∗2 , · · · , θ∗n) is a point with Crθ∗ ≥ 0.5.

Without loss of generality, let i be the index such that

Crθ∗ = min1≤k≤n

Crkθ∗k = Criθ∗i . (2.16)

We also immediately have

Crkθ∗k ≥ 0.5, k = 1, 2, · · · , n; (2.17)

Crkθ∗k+ supθk 6=θ∗k

Crkθk = 1, k = 1, 2, · · · , n; (2.18)

supθi 6=θ∗i

Criθi ≥ supθk 6=θ∗k

Crkθk, k = 1, 2, · · · , n; (2.19)


supθk 6=θ∗k

Crkθk ≤ 0.5, k = 1, · · · , n. (2.20)

It follows from (2.17) and (2.20) that

supθ 6=θ∗

Crθ = sup(θ1,θ2,··· ,θn) 6=(θ∗1 ,θ∗2 ,··· ,θ∗n)

min1≤k≤n

Crkθk

≥ supθi 6=θ∗i

min1≤k≤i−1

Crkθ∗k ∧ Criθi ∧ mini+1≤k≤n

Crkθ∗k

= supθi 6=θ∗i

Criθi.

We next suppose that

supθ 6=θ∗

Crθ > supθi 6=θ∗i

Criθi.

Then there is a point (θ′1, θ′2, · · · , θ′n) 6= (θ∗1 , θ

∗2 , · · · , θ∗n) such that

min1≤k≤n

Crkθ′k > supθi 6=θ∗i

Criθi.

Let j be one of the index such that θ′j 6= θ∗j . Then

Crjθ′j > supθi 6=θ∗i

Criθi.

That is,sup

θj 6=θ∗j

Crjθj > supθi 6=θ∗i

Criθi

which is in contradiction with (2.19). Thus

supθ 6=θ∗

Crθ = supθi 6=θ∗i

Criθi. (2.21)

It follows from (2.16), (2.18) and (2.21) that

Crθ∗+ supθ 6=θ∗

Crθ = Criθ∗i + supθi 6=θ∗i

Criθi = 1.

Thus Cr satisfies the credibility extension condition. It follows from the cred-ibility extension theorem that CrA is just the unique extension of Crθ.The theorem is proved.

Definition 2.3 Let (Θk,Pk,Crk), k = 1, 2, · · · , n be credibility spaces, Θ =Θ1×Θ2× · · · ×Θn and Cr = Cr1 ∧Cr2 ∧ · · · ∧Crn. Then (Θ,P,Cr) is calledthe product credibility space of (Θk,Pk,Crk), k = 1, 2, · · · , n.

Section 2.2 - Fuzzy Variables 63

Theorem 2.11 (Infinite Product Credibility Theorem) Suppose that Θk arenonempty sets, Crk the credibility measures on Pk, k = 1, 2, · · · , respectively.Let Θ = Θ1 ×Θ2 × · · · Then

CrA =

sup(θ1,θ2,··· )∈A

inf1≤k<∞

Crkθk,

if sup(θ1,θ2,··· )∈A

inf1≤k<∞

Crkθk < 0.5

1− sup(θ1,θ2,··· )∈Ac

inf1≤k<∞

Crkθk,

if sup(θ1,θ2,··· )∈A

inf1≤k<∞

Crkθk ≥ 0.5

is a credibility measure on P.

Proof: Like Theorem 2.10 except Cr(θ1, θ2, · · · ) = inf1≤k<∞ Crkθk.

Definition 2.4 Let (Θk,Pk,Crk), k = 1, 2, · · · be credibility spaces. DefineΘ = Θ1 × Θ2 × · · · and Cr = Cr1 ∧ Cr2 ∧ · · · Then (Θ,P,Cr) is called theinfinite product credibility space of (Θk,Pk,Crk), k = 1, 2, · · ·

2.2 Fuzzy Variables

Definition 2.5 A fuzzy variable is a (measurable) function from a credibilityspace (Θ,P,Cr) to the set of real numbers.

Example 2.7: Take (Θ,P,Cr) to be θ1, θ2 with Crθ1 = Crθ2 = 0.5.Then the function

ξ(θ) =

0, if θ = θ1

1, if θ = θ2

is a fuzzy variable.

Example 2.8: Take (Θ,P,Cr) to be the interval [0, 1] with Crθ = θ/2 foreach θ ∈ [0, 1]. Then the identity function ξ(θ) = θ is a fuzzy variable.

Example 2.9: A crisp number c may be regarded as a special fuzzy variable.In fact, it is the constant function ξ(θ) ≡ c on the credibility space (Θ,P,Cr).

Remark 2.3: Since a fuzzy variable ξ is a function on a credibility space,for any set B of real numbers, the set

ξ ∈ B =θ ∈ Θ

∣∣ ξ(θ) ∈ B (2.22)

is always an element in P. In other words, the fuzzy variable ξ is always ameasurable function and ξ ∈ B is always an event.


Definition 2.6 A fuzzy variable ξ is said to be(a) nonnegative if Crξ < 0 = 0;(b) positive if Crξ ≤ 0 = 0;(c) continuous if Crξ = x is a continuous function of x;(d) simple if there exists a finite sequence x1, x2, · · · , xm such that

Cr ξ 6= x1, ξ 6= x2, · · · , ξ 6= xm = 0; (2.23)


Cr ξ 6= x1, ξ 6= x2, · · · = 0. (2.24)

Definition 2.7 Let ξ1 and ξ2 be fuzzy variables defined on the credibilityspace (Θ,P,Cr). We say ξ1 = ξ2 if ξ1(θ) = ξ2(θ) for almost all θ ∈ Θ.

Fuzzy Vector

Definition 2.8 An n-dimensional fuzzy vector is defined as a function froma credibility space (Θ,P,Cr) to the set of n-dimensional real vectors.

Theorem 2.12 The vector (ξ1, ξ2, · · · , ξn) is a fuzzy vector if and only ifξ1, ξ2, · · · , ξn are fuzzy variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is a fuzzy vector. Thenξ1, ξ2, · · · , ξn are functions from Θ to <. Thus ξ1, ξ2, · · · , ξn are fuzzy vari-ables. Conversely, suppose that ξi are fuzzy variables defined on the cred-ibility spaces (Θi,Pi,Cri), i = 1, 2, · · · , n, respectively. It is clear that(ξ1, ξ2, · · · , ξn) is a function from the product credibility space (Θ,P,Cr)to <n, i.e.,

ξ(θ1, θ2, · · · , θn) = (ξ1(θ1), ξ2(θ2), · · · , ξn(θn))

for all (θ1, θ2, · · · , θn) ∈ Θ. Hence ξ = (ξ1, ξ2, · · · , ξn) is a fuzzy vector.

Fuzzy Arithmetic

In this subsection, we will suppose that all fuzzy variables are defined on acommon credibility space. Otherwise, we may embed them into the productcredibility space.

Definition 2.9 Let f : <n → < be a function, and ξ1, ξ2, · · · , ξn fuzzy vari-ables on the credibility space (Θ,P,Cr). Then ξ = f(ξ1, ξ2, · · · , ξn) is a fuzzyvariable defined as

ξ(θ) = f(ξ1(θ), ξ2(θ), · · · , ξn(θ)) (2.25)

for any θ ∈ Θ.

Section 2.3 - Membership Function 65

Example 2.10: Let ξ1 and ξ2 be fuzzy variables on the credibility space(Θ,P,Cr). Then their sum is

(ξ1 + ξ2)(θ) = ξ1(θ) + ξ2(θ), ∀θ ∈ Θ

and their product is

(ξ1 × ξ2)(θ) = ξ1(θ)× ξ2(θ), ∀θ ∈ Θ.

The reader may wonder whether ξ(θ1, θ2, · · · , θn) defined by (2.25) is afuzzy variable. The following theorem answers this question.

Theorem 2.13 Let ξ be an n-dimensional fuzzy vector, and f : <n → < afunction. Then f(ξ) is a fuzzy variable.

Proof: Since f(ξ) is a function from a credibility space to the set of realnumbers, it is a fuzzy variable.

2.3 Membership Function

Definition 2.10 Let ξ be a fuzzy variable defined on the credibility space(Θ,P,Cr). Then its membership function is derived from the credibility mea-sure by

µ(x) = (2Crξ = x) ∧ 1, x ∈ <. (2.26)

Membership function represents the degree that the fuzzy variable ξ takessome prescribed value. How do we determine membership functions? Thereare several methods reported in the past literature. Anyway, the membershipdegree µ(x) = 0 if x is an impossible point, and µ(x) = 1 if x is the mostpossible point that ξ takes.

Example 2.11: It is clear that a fuzzy variable has a unique membershipfunction. However, a membership function may produce multiple fuzzy vari-ables. For example, let Θ = θ1, θ2 and Crθ1 = Crθ2 = 0.5. Then(Θ,P,Cr) is a credibility space. We define

ξ1(θ) =

0, if θ = θ11, if θ = θ2,

ξ2(θ) =

1, if θ = θ10, if θ = θ2.

It is clear that both of them are fuzzy variables and have the same member-ship function, µ(x) ≡ 1 on x = 0 or 1.

Theorem 2.14 (Credibility Inversion Theorem) Let ξ be a fuzzy variablewith membership function µ. Then for any set B of real numbers, we have

Crξ ∈ B =12

(supx∈B

µ(x) + 1− supx∈Bc

µ(x)). (2.27)


Proof: If Crξ ∈ B ≤ 0.5, then by Axiom 2, we have Crξ = x ≤ 0.5 foreach x ∈ B. It follows from Axiom 4 that

Crξ ∈ B =12

(supx∈B

(2Crξ = x ∧ 1))

=12

supx∈B

µ(x). (2.28)

The self-duality of credibility measure implies that Crξ ∈ Bc ≥ 0.5 andsupx∈Bc Crξ = x ≥ 0.5, i.e.,

supx∈Bc

µ(x) = supx∈Bc

(2Crξ = x ∧ 1) = 1. (2.29)

It follows from (2.28) and (2.29) that (2.27) holds.If Crξ ∈ B ≥ 0.5, then Crξ ∈ Bc ≤ 0.5. It follows from the first case

that

Crξ ∈ B = 1− Crξ ∈ Bc = 1− 12

(sup

x∈Bc

µ(x) + 1− supx∈B

µ(x))

=12

(supx∈B


µ(x)).


Example 2.12: Let ξ be a fuzzy variable with membership function µ. Thenthe following equations follow immediately from Theorem 2.14:

Crξ = x =12

(µ(x) + 1− sup

y 6=xµ(y)

), ∀x ∈ <; (2.30)

Crξ ≤ x =12

(supy≤x

µ(y) + 1− supy>x

µ(y)), ∀x ∈ <; (2.31)

Crξ ≥ x =12

(supy≥x

µ(y) + 1− supy<x

µ(y)), ∀x ∈ <. (2.32)

Especially, if µ is a continuous function, then

Crξ = x =µ(x)

2, ∀x ∈ <. (2.33)

Theorem 2.15 (Sufficient and Necessary Condition for Membership Func-tion) A function µ : < → [0, 1] is a membership function if and only ifsupµ(x) = 1.

Proof: If µ is a membership function, then there exists a fuzzy variable ξwhose membership function is just µ, and

supx∈<

µ(x) = supx∈<

(2Crξ = x) ∧ 1.

Section 2.3 - Membership Function 67

If there is some point x ∈ < such that Crξ = x ≥ 0.5, then supµ(x) = 1.Otherwise, we have Crξ = x < 0.5 for each x ∈ <. It follows from Axiom 4that

supx∈<

µ(x) = supx∈<

(2Crξ = x) ∧ 1 = 2 supx∈<

Crξ = x = 2 (CrΘ ∧ 0.5) = 1.

Conversely, suppose that supµ(x) = 1. For each x ∈ <, we define

Crx =12

(µ(x) + 1− sup

y 6=xµ(y)

).

It is clear thatsupx∈<

Crx ≥ 12(1 + 1− 1) = 0.5.

For any x∗ ∈ < with Crx∗ ≥ 0.5, we have µ(x∗) = 1 and

Crx∗+ supy 6=x∗

Cry

=12

(µ(x∗) + 1− sup

y 6=x∗µ(y)

)+ sup

y 6=x∗

12

(µ(y) + 1− sup

z 6=yµ(z)

)

= 1− 12

supy 6=x∗

µ(y) +12

supy 6=x∗

µ(y) = 1.

Thus Crx satisfies the credibility extension condition, and has a uniqueextension to credibility measure on P(<) by using the credibility extensiontheorem. Now we define a fuzzy variable ξ as an identity function from thecredibility space (<,P(<),Cr) to <. Then the membership function of thefuzzy variable ξ is

(2Crξ = x) ∧ 1 =

(µ(x) + 1− sup

y 6=xµ(y)

)∧ 1 = µ(x)

for each x. The theorem is proved.

Remark 2.4: Theorem 2.15 states that the identity function is a universalfunction for any fuzzy variable by defining an appropriate credibility space.

Theorem 2.16 A fuzzy variable ξ with membership function µ is(a) nonnegative if and only if µ(x) = 0 for all x < 0;(b) positive if and only if µ(x) = 0 for all x ≤ 0;(c) simple if and only if µ takes nonzero values at a finite number of points;(d) discrete if and only if µ takes nonzero values at a countable set of points;(e) continuous if and only if µ is a continuous function.

Proof: The theorem is obvious since the membership function µ(x) =(2Crξ = x) ∧ 1 for each x ∈ <.


Some Special Membership Functions

By an equipossible fuzzy variable we mean the fuzzy variable fully determinedby the pair (a, b) of crisp numbers with a < b, whose membership function isgiven by

µ1(x) =

1, if a ≤ x ≤ b

0, otherwise.

By a triangular fuzzy variable we mean the fuzzy variable fully determinedby the triplet (a, b, c) of crisp numbers with a < b < c, whose membershipfunction is given by

µ2(x) =

x− a

b− a, if a ≤ x ≤ b

x− c

b− c, if b ≤ x ≤ c

0, otherwise.

By a trapezoidal fuzzy variable we mean the fuzzy variable fully determinedby the quadruplet (a, b, c, d) of crisp numbers with a < b < c < d, whosemembership function is given by

µ3(x) =

x− a

b− a, if a ≤ x ≤ b

1, if b ≤ x ≤ c

x− d

c− d, if c ≤ x ≤ d

0, otherwise.

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Figure 2.1: Membership Functions µ1, µ2 and µ3

Joint Membership Function

Definition 2.11 If ξ = (ξ1, ξ2, · · · , ξn) is a fuzzy vector on the credibilityspace (Θ,P,Cr). Then its joint membership function is derived from the

Section 2.4 - Credibility Distribution 69

credibility measure by

µ(x) = (2Crξ = x) ∧ 1, ∀x ∈ <n. (2.34)

Theorem 2.17 (Sufficient and Necessary Condition for Joint MembershipFunction) A function µ : <n → [0, 1] is a joint membership function if andonly if supµ(x) = 1.

Proof: Like Theorem 2.15.

2.4 Credibility Distribution

Definition 2.12 (Liu [124]) The credibility distribution Φ : < → [0, 1] of afuzzy variable ξ is defined by

Φ(x) = Crθ ∈ Θ

∣∣ ξ(θ) ≤ x. (2.35)

That is, Φ(x) is the credibility that the fuzzy variable ξ takes a value less thanor equal to x. Generally speaking, the credibility distribution Φ is neitherleft-continuous nor right-continuous.

Example 2.13: The credibility distribution of an equipossible fuzzy variable(a, b) is

Φ1(x) =

0, if x < a

1/2, if a ≤ x < b

1, if x ≥ b.

Especially, if ξ is an equipossible fuzzy variable on <, then Φ1(x) ≡ 1/2.

Example 2.14: The credibility distribution of a triangular fuzzy variable(a, b, c) is

Φ2(x) =

0, if x ≤ ax− a

2(b− a), if a ≤ x ≤ b

x+ c− 2b2(c− b)

, if b ≤ x ≤ c

1, if x ≥ c.


Example 2.15: The credibility distribution of a trapezoidal fuzzy variable(a, b, c, d) is

Φ3(x) =

0, if x ≤ ax− a

2(b− a), if a ≤ x ≤ b

12, if b ≤ x ≤ c

x+ d− 2c2(d− c)

, if c ≤ x ≤ d

1, if x ≥ d.

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Φ1(x)

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Figure 2.2: Credibility Distributions Φ1, Φ2 and Φ3

Theorem 2.18 Let ξ be a fuzzy variable with membership function µ. Thenits credibility distribution is

Φ(x) =12

(supy≤x

µ(y) + 1− supy>x

µ(y)), ∀x ∈ <. (2.36)

Proof: It follows from the credibility inversion theorem immediately.

Theorem 2.19 (Liu [129], Sufficient and Necessary Condition for Credibil-ity Distribution) A function Φ : < → [0, 1] is a credibility distribution if andonly if it is an increasing function with

limx→−∞

Φ(x) ≤ 0.5 ≤ limx→∞

Φ(x), (2.37)

limy↓x

Φ(y) = Φ(x) if limy↓x

Φ(y) > 0.5 or Φ(x) ≥ 0.5. (2.38)

Proof: It is obvious that a credibility distribution Φ is an increasing func-tion. The inequalities (2.37) follow from the credibility asymptotic theoremimmediately. Assume that x is a point at which limy↓x Φ(y) > 0.5. That is,

limy↓x

Crξ ≤ y > 0.5.


Since ξ ≤ y ↓ ξ ≤ x as y ↓ x, it follows from the credibility semicontinuitylaw that

Φ(y) = Crξ ≤ y ↓ Crξ ≤ x = Φ(x)

as y ↓ x. When x is a point at which Φ(x) ≥ 0.5, if limy↓x Φ(y) 6= Φ(x), thenwe have

limy↓x

Φ(y) > Φ(x) ≥ 0.5.

For this case, we have proved that limy↓x Φ(y) = Φ(x). Thus (2.37) and(2.38) are proved.

Conversely, if Φ : < → [0, 1] is an increasing function satisfying (2.37) and(2.38), then

µ(x) =

2Φ(x), if Φ(x) < 0.5

1, if limy↑x

Φ(y) < 0.5 ≤ Φ(x)

2− 2Φ(x), if 0.5 ≤ limy↑x

Φ(y)

(2.39)

takes values in [0, 1] and supµ(x) = 1. It follows from Theorem 2.15 thatthere is a fuzzy variable ξ whose membership function is just µ. Let us verifythat Φ is the credibility distribution of ξ, i.e., Crξ ≤ x = Φ(x) for each x.The argument breaks down into two cases. (i) If Φ(x) < 0.5, then we havesupy>x µ(y) = 1, and µ(y) = 2Φ(y) for each y with y ≤ x. Thus

Crξ ≤ x =12

(supy≤x

µ(y) + 1− supy>x

µ(y))

= supy≤x

Φ(y) = Φ(x).

(ii) If Φ(x) ≥ 0.5, then we have supy≤x µ(y) = 1 and Φ(y) ≥ Φ(x) ≥ 0.5 foreach y with y > x. Thus µ(y) = 2− 2Φ(y) and

Crξ ≤ x =12

(supy≤x

µ(y) + 1− supy>x

µ(y))

=12

(1 + 1− sup

y>x(2− 2Φ(y))

)= inf

y>xΦ(y) = lim

y↓xΦ(y) = Φ(x).


Example 2.16: Let a and b be two numbers with 0 ≤ a ≤ 0.5 ≤ b ≤ 1. Wedefine a fuzzy variable by the following membership function,

µ(x) =

2a, if x < 01, if x = 0

2− 2b, if x > 0.


Then its credibility distribution is

Φ(x) =

a, if x < 0b, if x ≥ 0.

Thus we havelim

x→−∞Φ(x) = a, lim

x→+∞Φ(x) = b.

Theorem 2.20 A fuzzy variable with credibility distribution Φ is(a) nonnegative if and only if Φ(x) = 0 for all x < 0;(b) positive if and only if Φ(x) = 0 for all x ≤ 0.

Proof: It follows immediately from the definition.

Theorem 2.21 Let ξ be a fuzzy variable. Then we have(a) if ξ is simple, then its credibility distribution is a simple function;(b) if ξ is discrete, then its credibility distribution is a step function;(c) if ξ is continuous on the real line <, then its credibility distribution is acontinuous function.

Proof: The parts (a) and (b) follow immediately from the definition. Thepart (c) follows from Theorem 2.18 and the continuity of the membershipfunction.

Example 2.17: However, the inverse of Theorem 2.21 is not true. Forexample, let ξ be a fuzzy variable whose membership function is

µ(x) =

x, if 0 ≤ x ≤ 11, otherwise.

Then its credibility distribution is Φ(x) ≡ 0.5. It is clear that Φ(x) is simpleand continuous. But the fuzzy variable ξ is neither simple nor continuous.

Definition 2.13 A continuous fuzzy variable is said to be (a) singular if itscredibility distribution is a singular function; (b) absolutely continuous if itscredibility distribution is absolutely continuous.

Definition 2.14 (Liu [124]) The credibility density function φ: < → [0,+∞)of a fuzzy variable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy, ∀x ∈ <, (2.40)

∫ +∞

−∞φ(y)dy = 1 (2.41)

where Φ is the credibility distribution of the fuzzy variable ξ.


Example 2.18: The credibility density function of a triangular fuzzy vari-able (a, b, c) is

φ(x) =

1

2(b− a), if a ≤ x ≤ b

12(c− b)

, if b ≤ x ≤ c

0, otherwise.

Example 2.19: The credibility density function of a trapezoidal fuzzy vari-able (a, b, c, d) is

φ(x) =

1

2(b− a), if a ≤ x ≤ b

12(d− c)

, if c ≤ x ≤ d

0, otherwise.

Example 2.20: The credibility density function of an equipossible fuzzyvariable (a, b) does not exist.

Example 2.21: The credibility density function does not necessarily existeven if the membership function is continuous and unimodal with a finitesupport. Let f be the Cantor function, and set

µ(x) =

f(x), if 0 ≤ x ≤ 1

f(2− x), if 1 < x ≤ 20, otherwise.

(2.42)

Then µ is a continuous and unimodal function with µ(1) = 1. Hence µis a membership function. However, its credibility distribution is not anabsolutely continuous function. Thus the credibility density function doesnot exist.

Theorem 2.22 Let ξ be a fuzzy variable whose credibility density functionφ exists. Then we have

Crξ ≤ x =∫ x

−∞φ(y)dy, Crξ ≥ x =

∫ +∞

x

φ(y)dy. (2.43)

Proof: The first part follows immediately from the definition. In addition,by the self-duality of credibility measure, we have

Crξ ≥ x = 1− Crξ < x =∫ +∞

−∞φ(y)dy −

∫ x

−∞φ(y)dy =

∫ +∞

x

φ(y)dy.



Example 2.22: Different from the random case, generally speaking,

Cra ≤ ξ ≤ b 6=∫ b

a

φ(y)dy.

Consider the trapezoidal fuzzy variable ξ = (1, 2, 3, 4). Then Cr2 ≤ ξ ≤3 = 0.5. However, it is obvious that φ(x) = 0 when 2 ≤ x ≤ 3 and∫ 3

2

φ(y)dy = 0 6= 0.5 = Cr2 ≤ ξ ≤ 3.

Joint Credibility Distribution

Definition 2.15 Let (ξ1, ξ2, · · · , ξn) be a fuzzy vector. Then the joint credi-bility distribution Φ : <n → [0, 1] is defined by

Φ(x1, x2, · · · , xn) = Crθ ∈ Θ

∣∣ ξ1(θ) ≤ x1, ξ2(θ) ≤ x2, · · · , ξn(θ) ≤ xn

.

Definition 2.16 The joint credibility density function φ : <n → [0,+∞) ofa fuzzy vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn

holds for all (x1, x2, · · · , xn) ∈ <n, and∫ +∞

−∞

∫ +∞

−∞· · ·∫ +∞

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn = 1

where Φ is the joint credibility distribution of the fuzzy vector (ξ1, ξ2, · · · , ξn).

2.5 Independence

The independence of fuzzy variables has been discussed by many authors fromdifferent angles, for example, Zadeh [248], Nahmias [163], Yager [231], Liu[129], Liu and Gao [148], and Li and Liu [99]. A lot of equivalence conditionsof independence are presented. Here we use the following condition.

Definition 2.17 (Liu and Gao [148]) The fuzzy variables ξ1, ξ2, · · · , ξm aresaid to be independent if

Cr

m⋂

i=1

ξi ∈ Bi

= min

1≤i≤mCr ξi ∈ Bi (2.44)

for any sets B1, B2, · · · , Bm of <.


Theorem 2.23 The fuzzy variables ξ1, ξ2, · · · , ξm are independent if andonly if

Cr

m⋃

i=1

ξi ∈ Bi

= max

1≤i≤mCr ξi ∈ Bi (2.45)

for any sets B1, B2, · · · , Bm of <.

Proof: It follows from the self-duality of credibility measure that ξ1, ξ2, · · · , ξmare independent if and only if

Cr

m⋃

i=1

ξi ∈ Bi

= 1− Cr

m⋂

i=1

ξi ∈ Bci

= 1− min

1≤i≤mCrξi ∈ Bc

i = max1≤i≤m

Cr ξi ∈ Bi .

Thus (2.45) is verified. The proof is complete.

Theorem 2.24 The fuzzy variables ξ1, ξ2, · · · , ξm are independent if andonly if

Cr

m⋂

i=1

ξi = xi

= min

1≤i≤mCr ξi = xi (2.46)

for any real numbers x1, x2, · · · , xm with Cr∩mi=1ξi = xi < 0.5.

Proof: If ξ1, ξ2, · · · , ξm are independent, then we have (2.46) immediatelyby taking Bi = xi for each i. Conversely, if Cr∩m

i=1ξi ∈ Bi ≥ 0.5, itfollows from Theorem 2.2 that (2.44) holds. Otherwise, we have Cr∩m

i=1ξi =xi < 0.5 for any real numbers xi ∈ Bi, i = 1, 2, · · · ,m, and

Cr

m⋂

i=1

ξi ∈ Bi

= Cr

⋃xi∈Bi,1≤i≤m

m⋂i=1

ξi = xi

= sup

xi∈Bi,1≤i≤mCr

m⋂

i=1

ξi = xi

= sup

xi∈Bi,1≤i≤mmin

1≤i≤mCrξi = xi

= min1≤i≤m

supxi∈Bi

Cr ξi = xi = min1≤i≤m

Cr ξi ∈ Bi .

Hence (2.44) is true, and ξ1, ξ2, · · · , ξm are independent. The theorem is thusproved.

Theorem 2.25 Let µi be membership functions of fuzzy variables ξi, i =1, 2, · · · ,m, respectively, and µ the joint membership function of fuzzy vector(ξ1, ξ2, · · · , ξm). Then the fuzzy variables ξ1, ξ2, · · · , ξm are independent ifand only if

µ(x1, x2, · · · , xm) = min1≤i≤m

µi(xi) (2.47)

for any real numbers x1, x2, · · · , xm.


Proof: Suppose that ξ1, ξ2, · · · , ξm are independent. It follows from Theo-rem 2.24 that

µ(x1, x2, · · · , xm) =

(2Cr

m⋂

i=1

ξi = xi

)∧ 1

=(

2 min1≤i≤m

Crξi = xi)∧ 1

= min1≤i≤m

(2Crξi = xi) ∧ 1 = min1≤i≤m

µi(xi).

Conversely, for any real numbers x1, x2, · · · , xm with Cr∩mi=1ξi = xi <

0.5, we have

Cr

m⋂

i=1

ξi = xi

=

12

(2Cr

m⋂

i=1

ξi = xi

)∧ 1

=12µ(x1, x2, · · · , xm) =

12

min1≤i≤m

µi(xi)

=12

(min

1≤i≤m(2Cr ξi = xi) ∧ 1

)= min

1≤i≤mCr ξi = xi .

It follows from Theorem 2.24 that ξ1, ξ2, · · · , ξm are independent. The theo-rem is proved.

Theorem 2.26 Let Φi be credibility distributions of fuzzy variables ξi, i =1, 2, · · · ,m, respectively, and Φ the joint credibility distribution of fuzzy vector(ξ1, ξ2, · · · , ξm). If ξ1, ξ2, · · · , ξm are independent, then we have

Φ(x1, x2, · · · , xm) = min1≤i≤m

Φi(xi) (2.48)

for any real numbers x1, x2, · · · , xm.

Proof: Since ξ1, ξ2, · · · , ξm are independent fuzzy variables, we have

Φ(x1, x2, · · · , xm) = Cr

m⋂

i=1

ξi ≤ xi

= min

1≤i≤mCrξi ≤ xi = min

1≤i≤mΦi(xi)

for any real numbers x1, x2, · · · , xm. The theorem is proved.

Example 2.23: However, the equation (2.48) does not imply that the fuzzyvariables are independent. For example, let ξ be a fuzzy variable with credi-bility distribution Φ. Then the joint credibility distribution Ψ of fuzzy vector(ξ, ξ) is

Ψ(x1, x2) = Crξ ≤ x1, ξ ≤ x2 = Crξ ≤ x1 ∧Crξ ≤ x2 = Φ(x1)∧Φ(x2)


for any real numbers x1 and x2. But, generally speaking, a fuzzy variable isnot independent with itself.

Theorem 2.27 Let ξ1, ξ2, · · · , ξm be independent fuzzy variables, and f1, f2,· · · , fn are real-valued functions. Then f1(ξ1), f2(ξ2), · · · , fm(ξm) are inde-pendent fuzzy variables.

Proof: For any sets B1, B2, · · · , Bm of <, we have

Cr

m⋂

i=1

fi(ξi) ∈ Bi

= Cr

m⋂

i=1

ξi ∈ f−1i (Bi)

= min

1≤i≤mCrξi ∈ f−1

i (Bi) = min1≤i≤m

Crfi(ξi) ∈ Bi.

Thus f1(ξ1), f2(ξ2), · · · , fm(ξm) are independent fuzzy variables.

Theorem 2.28 (Extension Principle of Zadeh) Let ξ1, ξ2, · · · , ξn be inde-pendent fuzzy variables with membership functions µ1, µ2, · · · , µn, respec-tively, and f : <n → < a function. Then the membership function µ ofξ = f(ξ1, ξ2, · · · , ξn) is derived from the membership functions µ1, µ2, · · · , µn

byµ(x) = sup

x=f(x1,x2,··· ,xn)

min1≤i≤n

µi(xi) (2.49)

for any x ∈ <. Here we set µ(x) = 0 if there are not real numbers x1, x2, · · · , xn

such that x = f(x1, x2, · · · , xn).

Proof: It follows from Definition 2.10 that the membership function of ξ =f(ξ1, ξ2, · · · , ξn) is

µ(x) = (2Cr f(ξ1, ξ2, · · · , ξn) = x) ∧ 1

=

2Cr

⋃x=f(x1,x2,··· ,xn)

ξ1 = x1, ξ2 = x2, · · · , ξn = xn

∧ 1

=

(2 sup

x=f(x1,x2,··· ,xn)

Crξ1 = x1, ξ2 = x2, · · · , ξn = xn

)∧ 1

=

(2 sup

x=f(x1,x2,··· ,xn)

min1≤k≤n

Crξi = xi

)∧ 1 (by independence)

= supx=f(x1,x2,··· ,xn)

min1≤k≤n

(2Crξi = xi) ∧ 1

= supx=f(x1,x2,··· ,xn)

min1≤i≤n

µi(xi).



Remark 2.5: The extension principle of Zadeh is only applicable to theoperations on independent fuzzy variables. In the past literature, the exten-sion principle is used as a postulate. However, it is treated as a theorem incredibility theory.

Example 2.24: The sum of independent equipossible fuzzy variables ξ =(a1, a2) and η = (b1, b2) is also an equipossible fuzzy variable, and

ξ + η = (a1 + b1, a2 + b2).

Their product is also an equipossible fuzzy variable, and

ξ · η =(

mina1≤x≤a2,b1≤y≤b2

xy, maxa1≤x≤a2,b1≤y≤b2

xy

).

Example 2.25: The sum of independent triangular fuzzy variables ξ =(a1, a2, a3) and η = (b1, b2, b3) is also a triangular fuzzy variable, and

ξ + η = (a1 + b1, a2 + b2, a3 + b3).

The product of a triangular fuzzy variable ξ = (a1, a2, a3) and a scalar numberλ is

λ · ξ =

(λa1, λa2, λa3), if λ ≥ 0

(λa3, λa2, λa1), if λ < 0.

That is, the product of a triangular fuzzy variable and a scalar number isalso a triangular fuzzy variable. However, the product of two triangular fuzzyvariables is not a triangular one.

Example 2.26: The sum of independent trapezoidal fuzzy variables ξ =(a1, a2, a3, a4) and η = (b1, b2, b3, b4) is also a trapezoidal fuzzy variable, and

ξ + η = (a1 + b1, a2 + b2, a3 + b3, a4 + b4).

The product of a trapezoidal fuzzy variable ξ = (a1, a2, a3, a4) and a scalarnumber λ is

λ · ξ =

(λa1, λa2, λa3, λa4), if λ ≥ 0

(λa4, λa3, λa2, λa1), if λ < 0.

That is, the product of a trapezoidal fuzzy variable and a scalar number isalso a trapezoidal fuzzy variable. However, the product of two trapezoidalfuzzy variables is not a trapezoidal one.

Example 2.27: Let ξ1, ξ2, · · · , ξn be independent fuzzy variables with mem-bership functions µ1, µ2, · · · , µn, respectively, and f : <n → < a function.Then for any set B of real numbers, the credibility Crf(ξ1, ξ2, · · · , ξn) ∈ Bis

12

(sup

f(x1,x2,··· ,xn)∈B

min1≤i≤n

µi(xi) + 1− supf(x1,x2,··· ,xn)∈Bc

min1≤i≤n

µi(xi)

).



Definition 2.18 (Liu [129]) The fuzzy variables ξ and η are said to be iden-tically distributed if

Crξ ∈ B = Crη ∈ B (2.50)

for any set B of <.

Theorem 2.29 The fuzzy variables ξ and η are identically distributed if andonly if ξ and η have the same membership function.

Proof: Let µ and ν be the membership functions of ξ and η, respectively. Ifξ and η are identically distributed fuzzy variables, then, for any x ∈ <, wehave

µ(x) = (2Crξ = x) ∧ 1 = (2Crη = x) ∧ 1 = ν(x).

Thus ξ and η have the same membership function.Conversely, if ξ and η have the same membership function, i.e., µ(x) ≡

ν(x), then, by using the credibility inversion theorem, we have

Crξ ∈ B =12

(supx∈B


µ(x))

=12

(supx∈B

ν(x) + 1− supx∈Bc

ν(x))

= Crη ∈ B

for any set B of <. Thus ξ and η are identically distributed fuzzy variables.

Theorem 2.30 The fuzzy variables ξ and η are identically distributed if andonly if Crξ = x = Crη = x for each x ∈ <.

Proof: If ξ and η are identically distributed fuzzy variables, then we im-mediately have Crξ = x = Crη = x for each x. Conversely, it followsfrom

µ(x) = (2Crξ = x) ∧ 1 = (2Crη = x) ∧ 1 = ν(x)

that ξ and η have the same membership function. Thus ξ and η are identicallydistributed fuzzy variables.

Theorem 2.31 If ξ and η are identically distributed fuzzy variables, then ξand η have the same credibility distribution.

Proof: If ξ and η are identically distributed fuzzy variables, then, for anyx ∈ <, we have Crξ ∈ (−∞, x] = Crη ∈ (−∞, x]. Thus ξ and η have thesame credibility distribution.

Example 2.28: The inverse of Theorem 2.31 is not true. We consider twofuzzy variables with the following membership functions,

µ(x) =

1.0, if x = 00.6, if x = 10.8, if x = 2,

ν(x) =

1.0, if x = 00.7, if x = 10.8, if x = 2.


It is easy to verify that ξ and η have the same credibility distribution,

Φ(x) =

0, if x < 0

0.6, if 0 ≤ x < 21, if x ≥ 2.

However, they are not identically distributed fuzzy variables.

Theorem 2.32 Let ξ and η be two fuzzy variables whose credibility densityfunctions exist. If ξ and η are identically distributed, then they have the samecredibility density function.

Proof: It follows from Theorem 2.31 that the fuzzy variables ξ and η have thesame credibility distribution. Hence they have the same credibility densityfunction.

2.7 Expected Value

There are many ways to define an expected value operator for fuzzy vari-ables. See, for example, Dubois and Prade [34], Heilpern [59], Campos andGonzalez [13], Gonzalez [53] and Yager [225][236]. The most general defini-tion of expected value operator of fuzzy variable was given by Liu and Liu[126]. This definition is applicable to not only continuous fuzzy variables butalso discrete ones.

Definition 2.19 (Liu and Liu [126]) Let ξ be a fuzzy variable. Then theexpected value of ξ is defined by

E[ξ] =∫ +∞

0

Crξ ≥ rdr −∫ 0

−∞Crξ ≤ rdr (2.51)


Example 2.29: Let ξ be the equipossible fuzzy variable (a, b). If a ≥ 0,then Crξ ≤ r ≡ 0 when r < 0, and

Crξ ≥ r =

1, if r ≤ a

0.5, if a < r ≤ b

0, if r > b,

E[ξ] =

(∫ a

0

1dr +∫ b

a

0.5dr +∫ +∞

b

0dr

)−∫ 0

−∞0dr =

a+ b

2.


If b ≤ 0, then Crξ ≥ r ≡ 0 when r > 0, and

Crξ ≤ r =

1, if r ≥ b

0.5, if a ≤ r < b

0, if r < a,

E[ξ] =∫ +∞

0

0dr −

(∫ a

−∞0dr +

∫ b

a

0.5dr +∫ 0

b

1dr

)=a+ b

2.

If a < 0 < b, then

Crξ ≥ r =

0.5, if 0 ≤ r ≤ b

0, if r > b,

Crξ ≤ r =

0, if r < a

0.5, if a ≤ r ≤ 0,

E[ξ] =

(∫ b

0

0.5dr +∫ +∞

b

0dr

)−(∫ a

−∞0dr +

∫ 0

a

0.5dr)

=a+ b

2.

Thus we always have the expected value (a+ b)/2.

Example 2.30: The triangular fuzzy variable ξ = (a, b, c) has an expectedvalue E[ξ] = (a+ 2b+ c)/4.

Example 2.31: The trapezoidal fuzzy variable ξ = (a, b, c, d) has an ex-pected value E[ξ] = (a+ b+ c+ d)/4.

Example 2.32: Let ξ be a continuous nonnegative fuzzy variable with mem-bership function µ. If µ is decreasing on [0,+∞), then Crξ ≥ x = µ(x)/2for any x > 0, and

E[ξ] =12

∫ +∞

0

µ(x)dx.

Example 2.33: Let ξ be a continuous fuzzy variable with membership func-tion µ. If its expected value exists, and there is a point x0 such that µ(x) isincreasing on (−∞, x0) and decreasing on (x0,+∞), then

E[ξ] = x0 +12

∫ +∞

x0

µ(x)dx− 12

∫ x0

−∞µ(x)dx.

Example 2.34: Let ξ be a fuzzy variable with membership function

µ(x) =

0, if x < 0x, if 0 ≤ x ≤ 11, if x > 1.


Then its expected value is +∞. If ξ is a fuzzy variable with membershipfunction

µ(x) =

1, if x < 0

1− x, if 0 ≤ x ≤ 10, if x > 1.

Then its expected value is −∞.

Example 2.35: The expected value may not exist for some fuzzy variables.For example, the fuzzy variable ξ with membership function

µ(x) =1

1 + |x|, ∀x ∈ <

does not have expected value because both of the integrals∫ +∞

0

Crξ ≥ rdr and∫ 0

−∞Crξ ≤ rdr

are infinite.

Example 2.36: The definition of expected value operator is also applicableto discrete case. Assume that ξ is a simple fuzzy variable whose membershipfunction is given by

µ(x) =

µ1, if x = x1

µ2, if x = x2

· · ·µm, if x = xm

(2.52)

where x1, x2, · · · , xm are distinct numbers. Note that µ1 ∨µ2 ∨ · · · ∨µm = 1.Definition 2.19 implies that the expected value of ξ is

E[ξ] =m∑

i=1

wixi (2.53)

where the weights are given by

wi =12

(max

1≤j≤mµj |xj ≤ xi − max

1≤j≤mµj |xj < xi

+ max1≤j≤m

µj |xj ≥ xi − max1≤j≤m

µj |xj > xi)

for i = 1, 2, · · · ,m. It is easy to verify that all wi ≥ 0 and the sum of allweights is just 1.

Example 2.37: Consider the fuzzy variable ξ defined by (2.52). Supposex1 < x2 < · · · < xm. Then the expected value is determined by (2.53) andthe weights are given by

wi =12

(max1≤j≤i

µj − max1≤j<i

µj + maxi≤j≤m

µj − maxi<j≤m

µj

)


for i = 1, 2, · · · ,m.

Example 2.38: Consider the fuzzy variable ξ defined by (2.52). Supposex1 < x2 < · · · < xm and there exists an index k with 1 < k < m such that

µ1 ≤ µ2 ≤ · · · ≤ µk and µk ≥ µk+1 ≥ · · · ≥ µm.

Note that µk ≡ 1. Then the expected value is determined by (2.53) and theweights are given by

wi =

µ1

2, if i = 1

µi − µi−1

2, if i = 2, 3, · · · , k − 1

1− µk−1 + µk+1

2, if i = k

µi − µi+1

2, if i = k + 1, k + 2, · · · ,m− 1

µm

2, if i = m.

Example 2.39: Consider the fuzzy variable ξ defined by (2.52). Supposex1 < x2 < · · · < xm and µ1 ≤ µ2 ≤ · · · ≤ µm (µm ≡ 1). Then the expectedvalue is determined by (2.53) and the weights are given by

wi =

µ1

2, if i = 1

µi − µi−1

2, if i = 2, 3, · · · ,m− 1

1− µm−1

2, if i = m.

Example 2.40: Consider the fuzzy variable ξ defined by (2.52). Supposex1 < x2 < · · · < xm and µ1 ≥ µ2 ≥ · · · ≥ µm (µ1 ≡ 1). Then the expectedvalue is determined by (2.53) and the weights are given by

wi =

1− µ2

2, if i = 1

µi − µi+1

2, if i = 2, 3, · · · ,m− 1

µm

2, if i = m.

Theorem 2.33 (Liu [124]) Let ξ be a fuzzy variable whose credibility densityfunction φ exists. If the Lebesgue integral∫ +∞

−∞xφ(x)dx



E[ξ] =∫ +∞

−∞xφ(x)dx. (2.54)

Proof: It follows from the definition of expected value operator and FubiniTheorem that

E[ξ] =∫ +∞

0


−∞Crξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.


Example 2.41: Let ξ be a fuzzy variable with credibility distribution Φ.Generally speaking,

E[ξ] 6=∫ +∞

−∞xdΦ(x).

For example, let ξ be a fuzzy variable with membership function

µ(x) =

0, if x < 0x, if 0 ≤ x ≤ 11, if x > 1.

Then E[ξ] = +∞. However∫ +∞

−∞xdΦ(x) =

146= +∞.

Theorem 2.34 (Liu [129]) Let ξ be a fuzzy variable with credibility distri-bution Φ. If

limx→−∞

Φ(x) = 0, limx→∞

Φ(x) = 1

and the Lebesgue-Stieltjes integral∫ +∞

−∞xdΦ(x)



E[ξ] =∫ +∞

−∞xdΦ(x). (2.55)


diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0


∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y


∫ y

−∞xdΦ(x) = 0.


y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y (1− Φ(y)) ≥ 0, for y > 0,

∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, for y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Crξ ≥ rdr


n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1)(xi+1 − xi) = y(Φ(y)− 1) → 0


0

Crξ ≥ rdr =∫ +∞

0

xdΦ(x).


−∫ 0

−∞Crξ ≤ rdr =

∫ 0

−∞xdΦ(x).

It follows that the equation (2.55) holds.


Linearity of Expected Value Operator

Theorem 2.35 (Liu and Liu [141]) Let ξ and η be independent fuzzy vari-ables with finite expected values. Then for any numbers a and b, we have

E[aξ + bη] = aE[ξ] + bE[η]. (2.56)

Proof: Step 1: We first prove that E[ξ + b] = E[ξ] + b for any real numberb. If b ≥ 0, we have

E[ξ + b] =∫ ∞

0

Crξ + b ≥ rdr −∫ 0

−∞Crξ + b ≤ rdr

=∫ ∞

0

Crξ ≥ r − bdr −∫ 0

−∞Crξ ≤ r − bdr

= E[ξ] +∫ b

0

(Crξ ≥ r − b+ Crξ < r − b) dr

= E[ξ] + b.


E[ξ + b] = E[ξ]−∫ 0

b

(Crξ ≥ r − b+ Crξ < r − b) dr = E[ξ] + b.

Step 2: We prove that E[aξ] = aE[ξ] for any real number a. If a = 0,then the equation E[aξ] = aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ ∞

0

Craξ ≥ rdr −∫ 0

−∞Craξ ≤ rdr

=∫ ∞

0

Crξ ≥ r

a

dr −

∫ 0

−∞Crξ ≤ r

a

dr

= a

∫ ∞

0

Crξ ≥ r

a

d( ra

)− a

∫ 0

−∞Crξ ≤ r

a

d( ra

)= aE[ξ].

If a < 0, we have

E[aξ] =∫ ∞

0

Craξ ≥ rdr −∫ 0

−∞Craξ ≤ rdr

=∫ ∞

0

Crξ ≤ r

a

dr −

∫ 0

−∞Crξ ≥ r

a

dr

= a

∫ ∞

0

Crξ ≥ r

a

d( ra

)− a

∫ 0

−∞Crξ ≤ r

a

d( ra

)= aE[ξ].


Step 3: We prove that E[ξ + η] = E[ξ] + E[η] when both ξ and η aresimple fuzzy variables with the following membership functions,

µ(x) =

µ1, if x = a1

µ2, if x = a2

· · ·µm, if x = am,

ν(x) =

ν1, if x = b1

ν2, if x = b2

· · ·νn, if x = bn.

Then ξ+η is also a simple fuzzy variable taking values ai+bj with membershipdegrees µi ∧ νj , i = 1, 2, · · · ,m, j = 1, 2, · · · , n, respectively. Now we define

w′i =12

(max

1≤k≤mµk|ak ≤ ai − max

1≤k≤mµk|ak < ai

+ max1≤k≤m

µk|ak ≥ ai − max1≤k≤m

µk|ak > ai),

w′′j =12

(max

1≤l≤nνl|bl ≤ bi − max

1≤l≤nνl|bl < bi

+ max1≤l≤n

νl|bl ≥ bi − max1≤l≤n

νl|bl > bi),

wij =12

(max

1≤k≤m,1≤l≤nµk ∧ νl|ak + bl ≤ ai + bj

− max1≤k≤m,1≤l≤n

µk ∧ νl|ak + bl < ai + bj

+ max1≤k≤m,1≤l≤n

µk ∧ νl|ak + bl ≥ ai + bj

− max1≤k≤m,1≤l≤n

µk ∧ νl|ak + bl > ai + bj)

for i = 1, 2, · · · ,m and j = 1, 2, · · · , n. It is also easy to verify that

w′i =n∑

j=1

wij , w′′j =m∑

i=1

wij

for i = 1, 2, · · · ,m and j = 1, 2, · · · , n. If ai, bj and ai + bj aresequences consisting of distinct elements, then

E[ξ] =m∑

i=1

aiw′i, E[η] =

n∑j=1

bjw′′j , E[ξ + η] =

m∑i=1

n∑j=1

(ai + bj)wij .

Thus E[ξ + η] = E[ξ] +E[η]. If not, we may give them a small perturbationsuch that they are distinct, and prove the linearity by letting the perturbationtend to zero.


Step 4: We prove that E[ξ + η] = E[ξ] + E[η] when ξ and η are fuzzyvariables such that

limy↑0

Crξ ≤ y ≤ 12≤ Crξ ≤ 0,

limy↑0

Crη ≤ y ≤ 12≤ Crη ≤ 0.

(2.57)

We define simple fuzzy variables ξi via credibility distributions as follows,

Φi(x) =

k − 12i

, ifk − 1

2i≤ Crξ ≤ x < k

2i, k = 1, 2, · · · , 2i−1

k

2i, if

k − 12i

≤ Crξ ≤ x < k

2i, k = 2i−1 + 1, · · · , 2i

1, if Crξ ≤ x = 1

for i = 1, 2, · · · Thus ξi is a sequence of simple fuzzy variables satisfying

Crξi ≤ r ↑ Crξ ≤ r, if r ≤ 0Crξi ≥ r ↑ Crξ ≥ r, if r ≥ 0

as i → ∞. Similarly, we define simple fuzzy variables ηi via credibilitydistributions as follows,

Ψi(x) =

k − 12i

, ifk − 1

2i≤ Crη ≤ x < k

2i, k = 1, 2, · · · , 2i−1

k

2i, if

k − 12i

≤ Crη ≤ x < k

2i, k = 2i−1 + 1, · · · , 2i

1, if Crη ≤ x = 1

for i = 1, 2, · · · Thus ηi is a sequence of simple fuzzy variables satisfying

Crηi ≤ r ↑ Crη ≤ r, if r ≤ 0Crηi ≥ r ↑ Crη ≥ r, if r ≥ 0

as i→∞. It is also clear that ξi+ηi is a sequence of simple fuzzy variables.Furthermore, when r ≤ 0, it follows from (2.57) that

limi→∞

Crξi + ηi ≤ r = limi→∞

supx≤0,y≤0,x+y≤r

Crξi ≤ x ∧ Crηi ≤ y

= supx≤0,y≤0,x+y≤r

limi→∞

Crξi ≤ x ∧ Crηi ≤ y

= supx≤0,y≤0,x+y≤r

Crξ ≤ x ∧ Crη ≤ y

= Crξ + η ≤ r.


That is,Crξi + ηi ≤ r ↑ Crξ + η ≤ r, if r ≤ 0.


Crξi + ηi ≥ r ↑ Crξ + η ≥ r, if r ≥ 0.

Since the expected values E[ξ] and E[η] exist, we have

E[ξi] =∫ +∞

0

Crξi ≥ rdr −∫ 0

−∞Crξi ≤ rdr

→∫ +∞

0


−∞Crξ ≤ rdr = E[ξ],

E[ηi] =∫ +∞

0

Crηi ≥ rdr −∫ 0

−∞Crηi ≤ rdr

→∫ +∞

0

Crη ≥ rdr −∫ 0

−∞Crη ≤ rdr = E[η],

E[ξi + ηi] =∫ +∞

0

Crξi + ηi ≥ rdr −∫ 0

−∞Crξi + ηi ≤ rdr

→∫ +∞

0

Crξ + η ≥ rdr −∫ 0

−∞Crξ + η ≤ rdr = E[ξ + η]

as i→∞. It follows from Step 3 that E[ξ + η] = E[ξ] + E[η].

Step 5: We prove that E[ξ + η] = E[ξ] + E[η] when ξ and η are arbi-trary fuzzy variables. Since they have finite expected values, there exist twonumbers c and d such that

limy↑0

Crξ + c ≤ y ≤ 12≤ Crξ + c ≤ 0,

limy↑0

Crη + d ≤ y ≤ 12≤ Crη + d ≤ 0.

It follows from Steps 1 and 4 that

E[ξ + η] = E[(ξ + c) + (η + d)− c− d]

= E[(ξ + c) + (η + d)]− c− d

= E[ξ + c] + E[η + d]− c− d

= E[ξ] + c+ E[η] + d− c− d

= E[ξ] + E[η].


Step 6: We prove that E[aξ + bη] = aE[ξ] + bE[η] for any real numbersa and b. In fact, the equation follows immediately from Steps 2 and 5. Thetheorem is proved.

Example 2.42: Theorem 2.35 does not hold if ξ and η are not independent.For example, take (Θ,P,Cr) to be θ1, θ2, θ3 with Crθ1 = 0.7, Crθ2 =0.3 and Crθ3 = 0.2. The fuzzy variables are defined by

ξ1(θ) =

1, if θ = θ1

0, if θ = θ2

2, if θ = θ3,

ξ2(θ) =

0, if θ = θ1

2, if θ = θ2

3, if θ = θ3.

Then we have

(ξ1 + ξ2)(θ) =

1, if θ = θ1

2, if θ = θ2

5, if θ = θ3.

Thus E[ξ1] = 0.9, E[ξ2] = 0.8, and E[ξ1 + ξ2] = 1.9. This fact implies that

E[ξ1 + ξ2] > E[ξ1] + E[ξ2].

If the fuzzy variables are defined by

η1(θ) =

0, if θ = θ1

1, if θ = θ2

2, if θ = θ3,

η2(θ) =

0, if θ = θ1

3, if θ = θ2

1, if θ = θ3.

Then we have

(η1 + η2)(θ) =

0, if θ = θ1

4, if θ = θ2

3, if θ = θ3.

Thus E[η1] = 0.5, E[η2] = 0.9, and E[η1 + η2] = 1.2. This fact implies that

E[η1 + η2] < E[η1] + E[η2].

Expected Value of Function of Fuzzy Variable

Let ξ be a fuzzy variable, and f : < → < a function. Then the expectedvalue of f(ξ) is

E[f(ξ)] =∫ +∞

0

Crf(ξ) ≥ rdr −∫ 0

−∞Crf(ξ) ≤ rdr.

For random case, it has been proved that the expected value E[f(ξ)] is theLebesgue-Stieltjes integral of f(x) with respect to the probability distribution


Φ of ξ if the integral exists. However, generally speaking, it is not true forfuzzy case.

Example 2.43: We consider a fuzzy variable ξ whose membership functionis given by

µ(x) =

0.6, if − 1 ≤ x < 01, if 0 ≤ x ≤ 10, otherwise.

Then the expected value E[ξ2] = 0.5. However, the credibility distributionof ξ is

Φ(x) =

0, if x < −1

0.3, if − 1 ≤ x < 00.5, if 0 ≤ x < 11, if x ≥ 1


−∞x2dΦ(x) = (−1)2 × 0.3 + 02 × 0.2 + 12 × 0.5 = 0.8 6= E[ξ2].

Remark 2.6: When f(x) is a monotone and continuous function, Zhu andJi [263] proved that

E[f(ξ)] =∫ +∞

−∞f(x)dΦ(x) (2.58)

where Φ is the credibility distribution of ξ.

Sum of a Fuzzy Number of Fuzzy Variables

Theorem 2.36 (Zhao and Liu [251]) Assume that ξi is a sequence of iidfuzzy variables, and n is a positive fuzzy integer (i.e., a fuzzy variable taking“positive integer” values) that is independent of the sequence ξi. Then wehave

E

[n∑

i=1

ξi

]= E [nξ1] . (2.59)

Proof: Since ξi is a sequence of iid fuzzy variables and n is independentof ξi, we have

Cr

n∑i=1

ξi ≥ r

= sup

n,x1+x2+···+xn≥rCrn = n ∧

(min

1≤i≤nCr ξi = xi

)≥ sup

nx≥rCrn = n ∧

(min

1≤i≤nCr ξi = x

)= sup

nx≥rCrn = n ∧ Crξ1 = x

= Cr nξ1 ≥ r .


On the other hand, for any given ε > 0, there exists an integer n and realnumbers x1, x2, · · · , xn with x1 + x2 + · · ·+ xn ≥ r such that

Cr

n∑

i=1

ξi ≥ r

− ε ≤ Crn = n ∧ Crξi = xi

for each i with 1 ≤ i ≤ n. Without loss of generality, we assume that nx1 ≥ r.Then we have

Cr

n∑

i=1

ξi ≥ r

− ε ≤ Crn = n ∧ Crξ1 = x1 ≤ Cr nξ1 ≥ r .

Letting ε→ 0, we get

Cr

n∑

i=1

ξi ≥ r

≤ Cr nξ1 ≥ r .

It follows that

Cr

n∑

i=1

ξi ≥ r

= Cr nξ1 ≥ r .

Similarly, the above identity still holds if the symbol “≥” is replaced with“≤”. Finally, by the definition of expected value operator, we have

E

[n∑

i=1

ξi

]=∫ +∞

0

Cr

n∑

i=1

ξi ≥ r

dr −

∫ 0

−∞Cr

n∑

i=1

ξi ≤ r

dr

=∫ +∞

0

Cr nξ1 ≥ rdr −∫ 0

−∞Cr nξ1 ≤ rdr = E [nξ1] .


2.8 Variance

Definition 2.20 (Liu and Liu [126]) Let ξ be a fuzzy variable with finiteexpected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

The variance of a fuzzy variable provides a measure of the spread of thedistribution around its expected value.

Example 2.44: Let ξ be an equipossible fuzzy variable (a, b). Then itsexpected value is e = (a+ b)/2, and for any positive number r, we have

Cr(ξ − e)2 ≥ r =

1/2, if r ≤ (b− a)2/4

0, if r > (b− a)2/4.


Thus the variance is

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr =∫ (b−a)2/4

0

12dr =

(b− a)2

8.

Example 2.45: Let ξ = (a, b, c) be a symmetric triangular fuzzy variable,i.e., b− a = c− b. Then its variance is V [ξ] = (c− a)2/24.

Example 2.46: Let ξ = (a, b, c, d) be a symmetric trapezoidal fuzzy variable,i.e., b − a = d − c. Then its variance is V [ξ] = ((d − a)2 + (d − a)(c − b) +(c− b)2)/24.

Example 2.47: A fuzzy variable ξ is called normally distributed if it has anormal membership function

µ(x) = 2(

1 + exp(π|x− e|√

6σ

))−1

, x ∈ <, σ > 0. (2.60)

The expected value is e and variance is σ2. Let ξ1 and ξ2 be independentlyand normally distributed fuzzy variables with expected values e1 and e2,variances σ2

1 and σ22 , respectively. Then for any real numbers a1 and a2, the

fuzzy variable a1ξ1 + a2ξ2 is also normally distributed with expected valuea1e1 + a2e2 and variance (|a1|σ1 + |a2|σ2)2.

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...............

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

.......................

...............

x

µ(x)

0

1

0.434

ee− σ e+ σ...................................................................

........................................................

...................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................

. . . . . . . . . . . . . . . . . . .

..

..

..

..

..

..

..

..

..

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..........

.........

Figure 2.3: Normal Membership Function

Theorem 2.37 If ξ is a fuzzy variable whose variance exists, a and b arereal numbers, then V [aξ + b] = a2V [ξ].


V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 2.38 Let ξ be a fuzzy variable with expected value e. Then V [ξ] =0 if and only if Crξ = e = 1.


Proof: If V [ξ] = 0, then E[(ξ − e)2] = 0. Note that

E[(ξ − e)2] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr

which implies Cr(ξ−e)2 ≥ r = 0 for any r > 0. Hence we have Cr(ξ−e)2 =0 = 1, i.e., Crξ = e = 1. Conversely, if Crξ = e = 1, then we haveCr(ξ − e)2 = 0 = 1 and Cr(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr = 0.


Let ξ be a fuzzy variable that takes values in [a, b], but whose membershipfunction is otherwise arbitrary. When its expected value is given, the maxi-mum variance theorem will provide the maximum variance of ξ, thus playingan important role in treating games against nature.

Theorem 2.39 (Li and Liu [104]) Let f be a convex function on [a, b], andξ a fuzzy variable that takes values in [a, b] and has expected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (2.61)

Proof: For each θ ∈ Θ, we have a ≤ ξ(θ) ≤ b and

ξ(θ) =b− ξ(θ)b− a

a+ξ(θ)− a

b− ab.


f(ξ(θ)) ≤ b− ξ(θ)b− a

f(a) +ξ(θ)− a

b− af(b).


Theorem 2.40 (Li and Liu [104], Maximum Variance Theorem) Let ξ be afuzzy variable that takes values in [a, b] and has expected value e. Then

V [ξ] ≤ (e− a)(b− e) (2.62)

and equality holds if the fuzzy variable ξ has membership function

µ(x) =

2(b− e)b− a

∧ 1, if x = a

2(e− a)b− a

∧ 1, if x = b.

(2.63)

Proof: It follows from Theorem 2.39 immediately by defining f(x) = (x−e)2.It is also easy to verify that the fuzzy variable determined by (2.63) hasvariance (e− a)(b− e). The theorem is proved.


2.9 Moments

Definition 2.21 (Liu [128]) Let ξ be a fuzzy variable, and k a positive num-ber. Then(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.


Example 2.48: A fuzzy variable ξ is called exponentially distributed if ithas an exponential membership function

µ(x) = 2(

1 + exp(

πx√6m

))−1

, x ≥ 0, m > 0. (2.64)

The expected value is (√

6m ln 2)/π and the second moment is m2. Let ξ1and ξ2 be independently and exponentially distributed fuzzy variables withsecond moments m2

1 and m22, respectively. Then for any positive real numbers

a1 and a2, the fuzzy variable a1ξ1+a2ξ2 is also exponentially distributed withsecond moment (a1m1 + a2m2)2.

.......................................................................................................................................................................................................................................................................................................................................................................................... ...............

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

.......................

...............

x

µ(x)

0

1

0.434

m

.........................................................................................................................................................................................................................................................................................................................................

. . . . . . . . . ..........

Figure 2.4: Exponential Membership Function

Theorem 2.41 Let ξ be a nonnegative fuzzy variable, and k a positive num-ber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1Crξ ≥ rdr. (2.65)


E[ξk] =∫ ∞

0

Crξk ≥ xdx =∫ ∞

0

Crξ ≥ rdrk = k

∫ ∞

0

rk−1Crξ ≥ rdr.



Theorem 2.42 (Li and Liu [104]) Let ξ be a fuzzy variable that takes val-ues in [a, b] and has expected value e. Then for any positive integer k, thekth absolute moment and kth absolute central moment satisfy the followinginequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (2.66)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (2.67)


2.10 Critical Values

In order to rank fuzzy variables, we may use two critical values: optimisticvalue and pessimistic value.

Definition 2.22 (Liu [124]) Let ξ be a fuzzy variable, and α ∈ (0, 1]. Then

ξsup(α) = supr∣∣ Cr ξ ≥ r ≥ α

(2.68)

is called the α-optimistic value to ξ, and

ξinf(α) = infr∣∣ Cr ξ ≤ r ≥ α

(2.69)

is called the α-pessimistic value to ξ.

This means that the fuzzy variable ξ will reach upwards of the α-optimisticvalue ξsup(α) with credibility α, and will be below the α-pessimistic valueξinf(α) with credibility α. In other words, the α-optimistic value ξsup(α) isthe supremum value that ξ achieves with credibility α, and the α-pessimisticvalue ξinf(α) is the infimum value that ξ achieves with credibility α.

Example 2.49: Let ξ be an equipossible fuzzy variable on (a, b). Then itsα-optimistic and α-pessimistic values are

ξsup(α) =

b, if α ≤ 0.5a, if α > 0.5,

ξinf(α) =

a, if α ≤ 0.5b, if α > 0.5.

Example 2.50: Let ξ = (a, b, c) be a triangular fuzzy variable. Then itsα-optimistic and α-pessimistic values are

ξsup(α) =

2αb+ (1− 2α)c, if α ≤ 0.5(2α− 1)a+ (2− 2α)b, if α > 0.5,


ξinf(α) =

(1− 2α)a+ 2αb, if α ≤ 0.5(2− 2α)b+ (2α− 1)c, if α > 0.5.

Example 2.51: Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable. Then itsα-optimistic and α-pessimistic values are

ξsup(α) =

2αc+ (1− 2α)d, if α ≤ 0.5(2α− 1)a+ (2− 2α)b, if α > 0.5,

ξinf(α) =

(1− 2α)a+ 2αb, if α ≤ 0.5(2− 2α)c+ (2α− 1)d, if α > 0.5.

Theorem 2.43 Let ξ be a fuzzy variable. If α > 0.5, then we have

Crξ ≤ ξinf(α) ≥ α, Crξ ≥ ξsup(α) ≥ α. (2.70)

Proof: It follows from the definition of α-pessimistic value that there existsa decreasing sequence xi such that Crξ ≤ xi ≥ α and xi ↓ ξinf(α) asi→∞. Since ξ ≤ xi ↓ ξ ≤ ξinf(α) and limi→∞ Crξ ≤ xi ≥ α > 0.5, itfollows from the credibility semicontinuity law that

Crξ ≤ ξinf(α) = limi→∞

Crξ ≤ xi ≥ α.

Similarly, there exists an increasing sequence xi such that Crξ ≥ xi ≥δ and xi ↑ ξsup(α) as i → ∞. Since ξ ≥ xi ↓ ξ ≥ ξsup(α) andlimi→∞ Crξ ≥ xi ≥ α > 0.5, it follows from the credibility semicontinuitylaw that

Crξ ≥ ξsup(α) = limi→∞

Crξ ≥ xi ≥ α.


Example 2.52: When α ≤ 0.5, it is possible that the inequalities

Crξ ≤ ξinf(α) < α, Crξ ≥ ξsup(α) < α

hold. Let ξ be an equipossible fuzzy variable on (−1, 1). It is clear thatξinf(0.5) = −1. However, Crξ ≤ ξinf(0.5) = 0 < 0.5. In addition,ξsup(0.5) = 1 and Crξ ≥ ξsup(0.5) = 0 < 0.5.

Theorem 2.44 Let ξ be a fuzzy variable. Then we have(a) ξinf(α) is an increasing and left-continuous function of α;(b) ξsup(α) is a decreasing and left-continuous function of α.

Proof: (a) It is easy to prove that ξinf(α) is an increasing function of α.Next, we prove the left-continuity of ξinf(α) with respect to α. Let αi bean arbitrary sequence of positive numbers such that αi ↑ α. Then ξinf(αi)


is an increasing sequence. If the limitation is equal to ξinf(α), then the left-continuity is proved. Otherwise, there exists a number z∗ such that

limi→∞


Thus Crξ ≤ z∗ ≥ αi for each i. Letting i → ∞, we get Crξ ≤ z∗ ≥ α.Hence z∗ ≥ ξinf(α). A contradiction proves the left-continuity of ξinf(α) withrespect to α. The part (b) may be proved similarly.

Theorem 2.45 Let ξ be a fuzzy variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).


1 ≥ Crξ < ξ(α)+ Crξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α). Part (b): Assume that ξinf(α) >ξsup(α). It follows from the definition of ξinf(α) that Crξ ≤ ξ(α) < α.Similarly, it follows from the definition of ξsup(α) that Crξ ≥ ξ(α) < α.Thus

1 ≤ Crξ ≤ ξ(α)+ Crξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is proved.

Theorem 2.46 Let ξ be a fuzzy variable. Then we have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Proof: If c = 0, then the part (a) is obviously valid. When c > 0, we have

(cξ)sup(α) = sup r | Crcξ ≥ r ≥ α= c sup r/c | Cr ξ ≥ r/c ≥ α= cξsup(α).

A similar way may prove that (cξ)inf(α) = cξinf(α).In order to prove the part (b), it suffices to verify that (−ξ)sup(α) =

−ξinf(α) and (−ξ)inf(α) = −ξsup(α). In fact, for any α ∈ (0, 1], we have

(−ξ)sup(α) = supr | Cr−ξ ≥ r ≥ α= − inf−r | Crξ ≤ −r ≥ α= −ξinf(α).



Theorem 2.47 Suppose that ξ and η are independent fuzzy variables. Thenfor any α ∈ (0, 1], we have

(ξ + η)sup(α) = ξsup(α) + ηsup(α), (ξ + η)inf(α) = ξinf(α) + ηinf(α),

(ξη)sup(α) = ξsup(α)ηsup(α), (ξη)inf(α) = ξinf(α)ηinf(α), if ξ ≥ 0, η ≥ 0,

(ξ ∨ η)sup(α) = ξsup(α) ∨ ηsup(α), (ξ ∨ η)inf(α) = ξinf(α) ∨ ηinf(α),

(ξ ∧ η)sup(α) = ξsup(α) ∧ ηsup(α), (ξ ∧ η)inf(α) = ξinf(α) ∧ ηinf(α).

Proof: For any given number ε > 0, since ξ and η are independent fuzzyvariables, we have

Crξ + η ≥ ξsup(α) + ηsup(α)− ε≥ Cr ξ ≥ ξsup(α)− ε/2 ∩ η ≥ ηsup(α)− ε/2= Crξ ≥ ξsup(α)− ε/2 ∧ Crη ≥ ηsup(α)− ε/2 ≥ α

which implies(ξ + η)sup(α) ≥ ξsup(α) + ηsup(α)− ε. (2.71)

On the other hand, by the independence, we have

Crξ + η ≥ ξsup(α) + ηsup(α) + ε≤ Cr ξ ≥ ξsup(α) + ε/2 ∪ η ≥ ηsup(α) + ε/2= Crξ ≥ ξsup(α) + ε/2 ∨ Crη ≥ ηsup(α) + ε/2 < α

which implies(ξ + η)sup(α) ≤ ξsup(α) + ηsup(α) + ε. (2.72)

It follows from (2.71) and (2.72) that

ξsup(α) + ηsup(α) + ε ≥ (ξ + η)sup(α) ≥ ξsup(α) + ηsup(α)− ε.

Letting ε → 0, we obtain (ξ + η)sup(α) = ξsup(α) + ηsup(α). The otherequalities may be proved similarly.

Example 2.53: The independence condition cannot be removed in Theo-rem 2.47. For example, let Θ = θ1, θ2, Crθ1 = Crθ2 = 1/2, and letfuzzy variables ξ and η be defined as

ξ(θ) =

0, if θ = θ1

1, if θ = θ2,η(θ) =

1, if θ = θ1

0, if θ = θ2.

However, (ξ + η)sup(0.6) = 1 6= 0 = ξsup(0.6) + ηsup(0.6).


2.11 Entropy

Fuzzy entropy is a measure of uncertainty and has been studied by manyresearchers such as De Luca and Termini [26], Kaufmann [75], Yager [223],Kosko [85], Pal and Pal [177], Bhandari and Pal [7], and Pal and Bezdek[181]. Those definitions of entropy characterize the uncertainty resultingprimarily from the linguistic vagueness rather than resulting from informationdeficiency, and vanishes when the fuzzy variable is an equipossible one.

In order to measure the uncertainty of fuzzy variables, Liu [131] suggestedthat an entropy of fuzzy variables should meet at least the following threebasic requirements:(i) minimum: the entropy of a crisp number is minimum, i.e., 0;(ii) maximum: the entropy of an equipossible fuzzy variable is maximum;(iii) universality: the entropy is applicable not only to finite and infinite casesbut also to discrete and continuous cases.

In order to meet those requirements, Li and Liu [95] provided a new def-inition of fuzzy entropy to characterize the uncertainty resulting from infor-mation deficiency which is caused by the impossibility to predict the specifiedvalue that a fuzzy variable takes.

Entropy of Discrete Fuzzy Variables

Definition 2.23 (Li and Liu [95]) Let ξ be a discrete fuzzy variable takingvalues in x1, x2, · · · . Then its entropy is defined by

H[ξ] =∞∑

i=1

S(Crξ = xi) (2.73)

where S(t) = −t ln t− (1− t) ln(1− t).

Remark 2.7: It is easy to verify that S(t) is a symmetric function aboutt = 0.5, strictly increases on the interval [0, 0.5], strictly decreases on theinterval [0.5, 1], and reaches its unique maximum ln 2 at t = 0.5.

Remark 2.8: It is clear that the entropy depends only on the number ofvalues and their credibilities and does not depend on the actual values thatthe fuzzy variable takes.

Example 2.54: Suppose that ξ is a discrete fuzzy variable taking values inx1, x2, · · · . If there exists some index k such that the membership functionµ(xk) = 1, and 0 otherwise, then its entropy H[ξ] = 0.

Example 2.55: Suppose that ξ is a simple fuzzy variable taking valuesin x1, x2, · · · , xn. If its membership function µ(x) ≡ 1, then its entropyH[ξ] = n ln 2.


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0 0.5 1t

S(t)

ln 2

..................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................

. . . . . . . . . . . . . . . . . . . ...........................

Figure 2.5: Function S(t) = −t ln t− (1− t) ln(1− t)

Theorem 2.48 Suppose that ξ is a discrete fuzzy variable taking values inx1, x2, · · · . Then

H[ξ] ≥ 0 (2.74)

and equality holds if and only if ξ is essentially a crisp number.

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifCrξ = xi = 0 or 1 for each i. That is, there exists one and only one indexk such that Crξ = xk = 1, i.e., ξ is essentially a crisp number.

This theorem states that the entropy of a fuzzy variable reaches its min-imum 0 when the fuzzy variable degenerates to a crisp number. In this case,there is no uncertainty.

Theorem 2.49 Suppose that ξ is a simple fuzzy variable taking values inx1, x2, · · · , xn. Then

H[ξ] ≤ n ln 2 (2.75)

and equality holds if and only if ξ is an equipossible fuzzy variable.

Proof: Since the function S(t) reaches its maximum ln 2 at t = 0.5, we have

H[ξ] =n∑

i=1

S(Crξ = xi) ≤ n ln 2

and equality holds if and only if Crξ = xi = 0.5, i.e., µ(xi) ≡ 1 for alli = 1, 2, · · · , n.

This theorem states that the entropy of a fuzzy variable reaches its max-imum when the fuzzy variable is an equipossible one. In this case, there isno preference among all the values that the fuzzy variable will take.


Entropy of Continuous Fuzzy Variables

Definition 2.24 (Li and Liu [95]) Let ξ be a continuous fuzzy variable.Then its entropy is defined by

H[ξ] =∫ +∞

−∞S(Crξ = x)dx (2.76)


For any continuous fuzzy variable ξ with membership function µ, we haveCrξ = x = µ(x)/2 for each x ∈ <. Thus

H[ξ] = −∫ +∞

−∞

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx. (2.77)

Example 2.56: Let ξ be an equipossible fuzzy variable (a, b). Then µ(x) = 1if a ≤ x ≤ b, and 0 otherwise. Thus its entropy is

H[ξ] = −∫ b

a

(12

ln12

+(

1− 12

)ln(

1− 12

))dx = (b− a) ln 2.

Example 2.57: Let ξ be a triangular fuzzy variable (a, b, c). Then itsentropy is H[ξ] = (c− a)/2.

Example 2.58: Let ξ be a trapezoidal fuzzy variable (a, b, c, d). Then itsentropy is H[ξ] = (d− a)/2 + (ln 2− 0.5)(c− b).

Example 2.59: Let ξ be an exponentially distributed fuzzy variable withsecond moment m2. Then its entropy is H[ξ] = πm/

√6.

Example 2.60: Let ξ be a normally distributed fuzzy variable with expectedvalue e and variance σ2. Then its entropy is H[ξ] =

√6πσ/3.

Theorem 2.50 Let ξ be a continuous fuzzy variable. Then H[ξ] > 0.

Proof: The positivity is clear. In addition, when a continuous fuzzy variabletends to a crisp number, its entropy tends to the minimum 0. However, acrisp number is not a continuous fuzzy variable.

Theorem 2.51 Let ξ be a continuous fuzzy variable taking values on theinterval [a, b]. Then

H[ξ] ≤ (b− a) ln 2 (2.78)

and equality holds if and only if ξ is an equipossible fuzzy variable (a, b).

Proof: The theorem follows from the fact that the function S(t) reaches itsmaximum ln 2 at t = 0.5.


Theorem 2.52 Let ξ and η be two continuous fuzzy variables with member-ship functions µ(x) and ν(x), respectively. If µ(x) ≤ ν(x) for any x ∈ <,then we have H[ξ] ≤ H[η].

Proof: Since µ(x) ≤ ν(x), we have S(µ(x)/2) ≤ S(ν(x)/2) for any x ∈ <.It follows that H[ξ] ≤ H[η].

Theorem 2.53 Let ξ be a continuous fuzzy variable. Then for any realnumbers a and b, we have H[aξ + b] = |a|H[ξ].

Proof: It follows from the definition of entropy that

H[aξ+b] =∫ +∞

−∞S(Craξ+b = x)dx = |a|

∫ +∞

−∞S(Crξ = y)dy = |a|H[ξ].

Maximum Entropy Principle

Given some constraints, for example, expected value and variance, there areusually multiple compatible membership functions. Which membership func-tion shall we take? The maximum entropy principle attempts to select themembership function that maximizes the value of entropy and satisfies theprescribed constraints.

Theorem 2.25 (Li and Liu [102]) Let ξ be a continuous nonnegative fuzzyvariable with finite second moment m2. Then

H[ξ] ≤ πm√6

(2.79)

and the equality holds if ξ is an exponentially distributed fuzzy variable withsecond moment m2.

Proof: Let µ be the membership function of ξ. Note that µ is a continuousfunction. The proof is based on the following two steps.

Step 1: Suppose that µ is a decreasing function on [0,+∞). For thiscase, we have Crξ ≥ x = µ(x)/2 for any x > 0. Thus the second moment

E[ξ2] =∫ +∞

0

Crξ2 ≥ xdx =∫ +∞

0

2xCrξ ≥ xdx =∫ +∞

0

xµ(x)dx.

The maximum entropy membership function µ should maximize the entropy

−∫ +∞

0

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx

subject to the moment constraint∫ +∞

0

xµ(x)dx = m2.


The Lagrangian is

L = −∫ +∞

0

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx

−λ(∫ +∞

0

xµ(x)dx−m2

).

The maximum entropy membership function meets Euler-Lagrange equation

12

lnµ(x)

2− 1

2ln(

1− µ(x)2

)+ λx = 0

and has the form µ(x) = 2 (1 + exp(2λx))−1. Substituting it into the momentconstraint, we get

µ∗(x) = 2(

1 + exp(

πx√6m

))−1

, x ≥ 0

which is just the exponential membership function with second moment m2,and the maximum entropy is H[ξ∗] = πm/

√6.

Step 2: Let ξ be a general fuzzy variable with second moment m2. Nowwe define a fuzzy variable ξ via membership function

µ(x) = supy≥x

µ(y), x ≥ 0.

Then µ is a decreasing function on [0,+∞), and

Crξ2 ≥ x =12

supy≥

√x

µ(y) =12

supy≥

√x

supz≥y

µ(z) =12

supz≥√

x

µ(z) ≤ Crξ2 ≥ x

for any x > 0. Thus we have

E[ξ2] =∫ +∞

0

Crξ2 ≥ xdx ≤∫ +∞

0

Crξ2 ≥ xdx = E[ξ2] = m2.

It follows from µ(x) ≤ µ(x) and Step 1 that

H[ξ] ≤ H[ξ] ≤π

√E[ξ2]√

6≤ πm√

6.

The theorem is thus proved.

Theorem 2.26 (Li and Liu [102]) Let ξ be a continuous fuzzy variable withfinite expected value e and variance σ2. Then

H[ξ] ≤√

6πσ3

(2.80)

and the equality holds if ξ is a normally distributed fuzzy variable with expectedvalue e and variance σ2.


Proof: Let µ be the continuous membership function of ξ. The proof isbased on the following two steps.

Step 1: Let µ(x) be a unimodal and symmetric function about x = e.For this case, the variance is

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ xdx =∫ +∞

0

Crξ − e ≥√xdx

=∫ +∞

e

2(x− e)Crξ ≥ xdx =∫ +∞

e

(x− e)µ(x)dx

and the entropy is

H[ξ] = −2∫ +∞

e

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx.

The maximum entropy membership function µ should maximize the entropysubject to the variance constraint. The Lagrangian is

L = −2∫ +∞

e

(µ(x)

2lnµ(x)

2+(

1− µ(x)2

)ln(

1− µ(x)2

))dx

−λ(∫ +∞

e

(x− e)µ(x)dx− σ2

).

The maximum entropy membership function meets Euler-Lagrange equation

lnµ(x)

2− ln

(1− µ(x)

2

)+ λ(x− e) = 0

and has the form µ(x) = 2 (1 + exp (λ(x− e)))−1. Substituting it into thevariance constraint, we get

µ∗(x) = 2(

1 + exp(π|x− e|√

6σ

))−1

, x ∈ <

which is just the normal membership function with expected value e andvariance σ2, and the maximum entropy is H[ξ∗] =

√6πσ/3.

Step 2: Let ξ be a general fuzzy variable with expected value e andvariance σ2. We define a fuzzy variable ξ by the membership function

µ(x) =

supy≤x

(µ(y) ∨ µ(2e− y)), if x ≤ e

supy≥x

(µ(y) ∨ µ(2e− y)) , if x > e.


It is easy to verify that µ(x) is a unimodal and symmetric function aboutx = e. Furthermore,

Cr

(ξ − e)2 ≥ r

=12

supx≥e+

√r

µ(x) =12

supx≥e+

√r

supy≥x

(µ(y) ∨ µ(2e− y))

=12

supy≥e+

√r

(µ(y) ∨ µ(2e− y)) =12

sup(y−e)2≥r

µ(y)

≤ Cr(ξ − e)2 ≥ r

for any r > 0. Thus

V [ξ] =∫ +∞

0

Cr(ξ − e)2 ≥ rdr ≤∫ +∞

0

Cr(ξ − e)2 ≥ rdr = σ2.

It follows from µ(x) ≤ µ(x) and Step 1 that

H[ξ] ≤ H[ξ] ≤√

6π√V [ξ]

3≤√

6πσ3

.

The proof is complete.

2.12 Distance

Distance between fuzzy variables has been defined in many ways, for exam-ple, Hausdorff distance (Puri and Ralescu [192], Klement et. al. [81]), andHamming distance (Kacprzyk [72]). However, those definitions have no iden-tification property. In order to overcome this shortage, Liu [129] proposed adefinition of distance as follows.

Definition 2.27 (Liu [129]) The distance between fuzzy variables ξ and η isdefined as

d(ξ, η) = E[|ξ − η|]. (2.81)

Example 2.61: Let ξ and η be equipossible fuzzy variables (a1, b1) and(a2, b2), respectively, and (a1, b1)∩(a2, b2) = ∅. Then |ξ−η| is an equipossiblefuzzy variable on the interval with endpoints |a1 − b2| and |b1 − a2|. Thusthe distance between ξ and η is the expected value of |ξ − η|, i.e.,

d(ξ, η) =12

(|a1 − b2|+ |b1 − a2|) .

Example 2.62: Let ξ = (a1, b1, c1) and η = (a2, b2, c2) be triangular fuzzyvariables such that (a1, c1) ∩ (a2, c2) = ∅. Then

d(ξ, η) =14

(|a1 − c2|+ 2|b1 − b2|+ |c1 − a2|) .


Example 2.63: Let ξ = (a1, b1, c1, d1) and η = (a2, b2, c2, d2) be trapezoidalfuzzy variables such that (a1, d1) ∩ (a2, d2) = ∅. Then

d(ξ, η) =14

(|a1 − d2|+ |b1 − c2|+ |c1 − b2|+ |d1 − a2|) .

Theorem 2.54 (Li and Liu [105]) Let ξ, η, τ be fuzzy variables, and let d(·, ·)be the distance. Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ) + 2d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.Now we prove the part (d). It follows from the credibility subadditivitytheorem that

d(ξ, η) =∫ +∞

0

Cr |ξ − η| ≥ rdr

≤∫ +∞

0

Cr |ξ − τ |+ |τ − η| ≥ rdr

≤∫ +∞

0

Cr |ξ − τ | ≥ r/2 ∪ |τ − η| ≥ r/2dr

≤∫ +∞

0

(Cr|ξ − τ | ≥ r/2+ Cr|τ − η| ≥ r/2) dr

=∫ +∞

0

Cr|ξ − τ | ≥ r/2dr +∫ +∞

0

Cr|τ − η| ≥ r/2dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ) + 2d(τ, η).

Example 2.64: Let Θ = θ1, θ2, θ3 and Crθi = 1/2 for i = 1, 2, 3. Wedefine fuzzy variables ξ, η and τ as follows,

ξ(θ) =

1, if θ 6= θ3

0, otherwise,η(θ) =

−1, if θ 6= θ1

0, otherwise,τ(θ) ≡ 0.

It is easy to verify that d(ξ, τ) = d(τ, η) = 1/2 and d(ξ, η) = 3/2. Thus

d(ξ, η) =32(d(ξ, τ) + d(τ, η)).

2.13 Inequalities

There are several useful inequalities for random variable, such as Markovinequality, Chebyshev inequality, Holder’s inequality, Minkowski inequality,


and Jensen’s inequality. This section introduces the analogous inequalitiesfor fuzzy variable.

Theorem 2.55 (Liu [128]) Let ξ be a fuzzy variable, and f a nonnegativefunction. If f is even and increasing on [0,∞), then for any given numbert > 0, we have

Cr|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (2.82)

Proof: It is clear that Cr|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Crf(ξ) ≥ rdr

=∫ +∞

0

Cr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

Cr|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr · Cr|ξ| ≥ f−1(f(t))

= f(t) · Cr|ξ| ≥ t


Theorem 2.56 (Liu [128], Markov Inequality) Let ξ be a fuzzy variable.Then for any given numbers t > 0 and p > 0, we have

Cr|ξ| ≥ t ≤ E[|ξ|p]tp

. (2.83)


Theorem 2.57 (Liu [128], Chebyshev Inequality) Let ξ be a fuzzy variablewhose variance V [ξ] exists. Then for any given number t > 0, we have

Cr |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (2.84)

Proof: It is a special case of Theorem 2.55 when the fuzzy variable ξ isreplaced with ξ − E[ξ], and f(x) = x2.

Theorem 2.58 (Liu [128], Holder’s Inequality) Let p and q be two positivereal numbers with 1/p+1/q = 1, and let ξ and η be independent fuzzy variableswith E[|ξ|p] <∞ and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (2.85)









E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).


Theorem 2.59 (Liu [128], Minkowski Inequality) Let p be a real numberwith p ≥ 1, and let ξ and η be independent fuzzy variables with E[|ξ|p] < ∞and E[|η|p] <∞. Then we have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (2.86)


√x + p







E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Theorem 2.60 (Liu [129], Jensen’s Inequality) Let ξ be a fuzzy variable,and f : < → < a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (2.87)




f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).


E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0



This section discusses some convergence concepts of fuzzy sequence: conver-gence almost surely (a.s.), convergence in credibility, convergence in mean,and convergence in distribution.

Table 2.1: Relations among Convergence Concepts

Convergence Almost SurelyConvergence

→Convergence

in Mean in Credibility Convergence in Distribution

Definition 2.28 (Liu [128]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variables de-fined on the credibility space (Θ,P,Cr). The sequence ξi is said to be con-vergent a.s. to ξ if and only if there exists an event A with CrA = 1 suchthat

limi→∞

|ξi(θ)− ξ(θ)| = 0 (2.88)

for every θ ∈ A. In that case we write ξi → ξ, a.s.

Definition 2.29 (Liu [128]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variables de-fined on the credibility space (Θ,P,Cr). We say that the sequence ξi con-verges in credibility to ξ if

limi→∞

Cr |ξi − ξ| ≥ ε = 0 (2.89)

for every ε > 0.

Definition 2.30 (Liu [128]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variableswith finite expected values defined on the credibility space (Θ,P,Cr). Wesay that the sequence ξi converges in mean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (2.90)



limi→∞

E[|ξi − ξ|2] = 0. (2.91)

Definition 2.31 (Liu [128]) Suppose that Φ,Φ1,Φ2, · · · are the credibilitydistributions of fuzzy variables ξ, ξ1, ξ2, · · · , respectively. We say that ξiconverges in distribution to ξ if Φi → Φ at any continuity point of Φ.

Convergence in Mean vs. Convergence in Credibility

Theorem 2.61 (Liu [128]) Suppose that ξ, ξ1, ξ2, · · · are fuzzy variables de-fined on the credibility space (Θ,P,Cr). If the sequence ξi converges inmean to ξ, then ξi converges in credibility to ξ.

Proof: It follows from Theorem 2.56 that, for any given number ε > 0,

Cr |ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in credibility to ξ.

Example 2.65: Convergence in credibility does not imply convergence inmean. For example, take (Θ,P,Cr) to be θ1, θ2, · · · with Crθ1 = 1/2 andCrθj = 1/j for j = 2, 3, · · · The fuzzy variables are defined by

ξi(θj) =

i, if j = i

0, otherwise


Cr |ξi − ξ| ≥ ε =1i→ 0.

That is, the sequence ξi converges in credibility to ξ. However,

E [|ξi − ξ|] ≡ 1 6→ 0.


Convergence Almost Surely vs. Convergence in Credibility

Example 2.66: Convergence a.s. does not imply convergence in credibility.For example, take (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j + 1) forj = 1, 2, · · · The fuzzy variables are defined by

ξi(θj) =

i, if j = i

0, otherwise


for i = 1, 2, · · · and ξ = 0. Then the sequence ξi converges a.s. to ξ.However, for any small number ε > 0, we have

Cr |ξi − ξ| ≥ ε =i

2i+ 1→ 1

2.

That is, the sequence ξi does not converge in credibility to ξ.

Theorem 2.62 (Wang and Liu [221]) Suppose that ξ, ξ1, ξ2, · · · are fuzzyvariables defined on the credibility space (Θ,P,Cr). If the sequence ξi con-verges in credibility to ξ, then ξi converges a.s. to ξ.

Proof: If ξi does not converge a.s. to ξ, then there exists an elementθ∗ ∈ Θ with Crθ∗ > 0 such that ξi(θ∗) 6→ ξ(θ∗) as i→∞. In other words,there exists a small number ε > 0 and a subsequence ξik

(θ∗) such that|ξik

(θ∗)− ξ(θ∗)| ≥ ε for any k. Since credibility measure is an increasing setfunction, we have

Cr |ξik− ξ| ≥ ε ≥ Crθ∗ > 0

for any k. It follows that ξi does not converge in credibility to ξ. Acontradiction proves the theorem.

Convergence in Credibility vs. Convergence in Distribution

Theorem 2.63 (Wang and Liu [221]) Suppose that ξ, ξ1, ξ2, · · · are fuzzyvariables. If the sequence ξi converges in credibility to ξ, then ξi con-verges in distribution to ξ.

Proof: Let x be any given continuity point of the distribution Φ. On theone hand, for any y > x, we have

ξi ≤ x = ξi ≤ x, ξ ≤ y ∪ ξi ≤ x, ξ > y ⊂ ξ ≤ y ∪ |ξi − ξ| ≥ y − x.

It follows from the credibility subadditivity theorem that

Φi(x) ≤ Φ(y) + Cr|ξi − ξ| ≥ y − x.

Since ξi converges in credibility to ξ, we have Cr|ξi − ξ| ≥ y − x → 0.Thus we obtain lim supi→∞ Φi(x) ≤ Φ(y) for any y > x. Letting y → x, weget

lim supi→∞

Φi(x) ≤ Φ(x). (2.92)


ξ ≤ z = ξ ≤ z, ξi ≤ x ∪ ξ ≤ z, ξi > x ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) + Cr|ξi − ξ| ≥ x− z.


Since Cr|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞ Φi(x) for anyz < x. Letting z → x, we get


Φi(x). (2.93)


Example 2.67: Convergence in distribution does not imply convergencein credibility. For example, take (Θ,P,Cr) to be θ1, θ2 with Crθ1 =Crθ2 = 1/2, and define

ξ(θ) =−1, if θ = θ11, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · Then ξi and ξ are identically dis-tributed. Thus ξi converges in distribution to ξ. But, for any small numberε > 0, we have Cr|ξi − ξ| > ε = CrΘ = 1. That is, the sequence ξidoes not converge in credibility to ξ.

Convergence Almost Surely vs. Convergence in Distribution

Example 2.68: Convergence in distribution does not imply convergence a.s.For example, take (Θ,P,Cr) to be θ1, θ2 with Crθ1 = Crθ2 = 1/2, anddefine

ξ(θ) =−1, if θ = θ11, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · Then ξi converges in distributionto ξ. However, ξi does not converge a.s. to ξ.

Example 2.69: Convergence a.s. does not imply convergence in distribution.For example, take (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j + 1) forj = 1, 2, · · · The fuzzy variables are defined by

ξi(θj) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ = 0. Then the sequence ξi converges a.s. to ξ.However, the credibility distributions of ξi are

Φi(x) =

0, if x < 0

(i+ 1)/(2i+ 1), if 0 ≤ x < i

1, if x ≥ i,

i = 1, 2, · · · , respectively. The credibility distribution of ξ is

Φ(x) =

0, if x < 01, if x ≥ 0.


It is clear that Φi(x) 6→ Φ(x) at x > 0. That is, the sequence ξi does notconverge in distribution to ξ.

2.15 Conditional Credibility

We now consider the credibility of an event A after it has been learned thatsome other event B has occurred. This new credibility of A is called theconditional credibility of A given B.

The first problem is whether the conditional credibility is determineduniquely and completely. The answer is negative. It is doomed to failure todefine an unalterable and widely accepted conditional credibility. For thisreason, it is appropriate to speak of a certain person’s subjective conditionalcredibility, rather than to speak of the true conditional credibility.

In order to define a conditional credibility measure CrA|B, at first wehave to enlarge CrA ∩ B because CrA ∩ B < 1 for all events wheneverCrB < 1. It seems that we have no alternative but to divide CrA∩B byCrB. Unfortunately, CrA∩B/CrB is not always a credibility measure.However, the value CrA|B should not be greater than CrA ∩ B/CrB(otherwise the normality will be lost), i.e.,

CrA|B ≤ CrA ∩BCrB

. (2.94)

On the other hand, in order to preserve the self-duality, we should have

CrA|B = 1− CrAc|B ≥ 1− CrAc ∩BCrB

. (2.95)

Furthermore, since (A ∩B) ∪ (Ac ∩B) = B, we have CrB ≤ CrA ∩B+CrAc ∩B by using the credibility subadditivity theorem. Thus

0 ≤ 1− CrAc ∩BCrB

≤ CrA ∩BCrB

≤ 1. (2.96)

Hence any numbers between 1−CrAc ∩B/CrB and CrA∩B/CrBare reasonable values that the conditional credibility may take. Based on themaximum uncertainty principle, we have the following conditional credibilitymeasure.

Definition 2.32 (Liu [132]) Let (Θ,P,Cr) be a credibility space, and A,B ∈P. Then the conditional credibility measure of A given B is defined by

CrA|B =

CrA ∩BCrB

, ifCrA ∩B

CrB< 0.5

1− CrAc ∩BCrB

, ifCrAc ∩B

CrB< 0.5

0.5, otherwise

(2.97)

Section 2.15 - Conditional Credibility 115

provided that CrB > 0.

It follows immediately from the definition of conditional credibility that

1− CrAc ∩BCrB

≤ CrA|B ≤ CrA ∩BCrB

. (2.98)

Furthermore, the value of CrA|B takes values as close to 0.5 as possiblein the interval. In other words, it accords with the maximum uncertaintyprinciple.

Theorem 2.64 (Liu [132]) Let (Θ,P,Cr) be a credibility space, and B anevent with CrB > 0. Then Cr·|B defined by (2.97) is a credibility mea-sure, and (Θ,P,Cr·|B) is a credibility space.

Proof: It is sufficient to prove that Cr·|B satisfies the normality, mono-tonicity, self-duality and maximality axioms. At first, it satisfies the normal-ity axiom, i.e.,

CrΘ|B = 1− CrΘc ∩BCrB

= 1− Cr∅CrB

= 1.

For any events A1 and A2 with A1 ⊂ A2, if

CrA1 ∩BCrB

≤ CrA2 ∩BCrB

< 0.5,

then

CrA1|B =CrA1 ∩B

CrB≤ CrA2 ∩B

CrB= CrA2|B.

IfCrA1 ∩B

CrB≤ 0.5 ≤ CrA2 ∩B

CrB,

then CrA1|B ≤ 0.5 ≤ CrA2|B. If

0.5 <CrA1 ∩B

CrB≤ CrA2 ∩B

CrB,

then we have

CrA1|B =(1− CrAc

1 ∩BCrB

)∨0.5 ≤

(1− CrAc

2 ∩BCrB

)∨0.5 = CrA2|B.

This means that Cr·|B satisfies the monotonicity axiom. For any event A,if

CrA ∩BCrB

≥ 0.5,CrAc ∩B

CrB≥ 0.5,


then we have CrA|B+ CrAc|B = 0.5 + 0.5 = 1 immediately. Otherwise,without loss of generality, suppose

CrA ∩BCrB

< 0.5 <CrAc ∩B

CrB,

then we have

CrA|B+ CrAc|B =CrA ∩B

CrB+(

1− CrA ∩BCrB

)= 1.

That is, Cr·|B satisfies the self-duality axiom. Finally, for any events Aiwith supi CrAi|B < 0.5, we have supi CrAi ∩B < 0.5 and

supi

CrAi|B =supi CrAi ∩B

CrB=

Cr∪iAi ∩BCrB

= Cr∪iAi|B.

Thus Cr·|B satisfies the maximality axiom. Hence Cr·|B is a credibilitymeasure. Furthermore, (Θ,P,Cr·|B) is a credibility space.

Example 2.70: Let ξ be a fuzzy variable, and X a set of real numbers suchthat Crξ ∈ X > 0. Then for any x ∈ X, the conditional credibility ofξ = x given ξ ∈ X is

Cr ξ = x|ξ ∈ X =

Crξ = xCrξ ∈ X

, ifCrξ = xCrξ ∈ X

< 0.5

1− Crξ 6= x, ξ ∈ XCrξ ∈ X

, ifCrξ 6= x, ξ ∈ X

Crξ ∈ X< 0.5

0.5, otherwise.

Example 2.71: Let ξ and η be two fuzzy variables, and Y a set of realnumbers such that Crη ∈ Y > 0. Then we have

Cr ξ = x|η ∈ Y =

Crξ = x, η ∈ Y Crη ∈ Y

, ifCrξ = x, η ∈ Y

Crη ∈ Y < 0.5

1− Crξ 6= x, η ∈ Y Crη ∈ Y

, ifCrξ 6= x, η ∈ Y

Crη ∈ Y < 0.5

0.5, otherwise.

Definition 2.33 (Liu [132]) The conditional membership function of a fuzzyvariable ξ given B is defined by

µ(x|B) = (2Crξ = x|B) ∧ 1, x ∈ < (2.99)


Section 2.15 - Conditional Credibility 117

Example 2.72: Let ξ be a fuzzy variable with membership function µ(x),and X a set of real numbers such that µ(x) > 0 for some x ∈ X. Then theconditional membership function of ξ given ξ ∈ X is

µ(x|X) =

2µ(x)supx∈X

µ(x)∧ 1, if sup

x∈Xµ(x) < 1

2µ(x)2− sup

x∈Xc

µ(x)∧ 1, if sup

x∈Xµ(x) = 1

(2.100)

for x ∈ X. Please mention that µ(x|X) ≡ 0 if x 6∈ X.

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µ(x|X)

0

1

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0

1

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µ(x|X)

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..

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..

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..

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Figure 2.6: Conditional Membership Function µ(x|X)

Example 2.73: Let ξ and η be two fuzzy variables with joint member-ship function µ(x, y), and Y a set of real numbers. Then the conditionalmembership function of ξ given η ∈ Y is

µ(x|Y ) =

2 supy∈Y

µ(x, y)

supx∈<,y∈Y

µ(x, y)∧ 1, if sup

x∈<,y∈Yµ(x, y) < 1

2 supy∈Y

µ(x, y)

2− supx∈<,y∈Y c


x∈<,y∈Yµ(x, y) = 1

(2.101)

provided that µ(x, y) > 0 for some x ∈ < and y ∈ Y . Especially, theconditional membership function of ξ given η = y is

µ(x|y) =

2µ(x, y)supx∈<


x∈<µ(x, y) < 1

2µ(x, y)2− sup

x∈<,z 6=yµ(x, z)

∧ 1, if supx∈<

µ(x, y) = 1


provided that µ(x, y) > 0 for some x ∈ <.

Definition 2.34 (Liu [132]) The conditional credibility distribution Φ: < →[0, 1] of a fuzzy variable ξ given B is defined by

Φ(x|B) = Cr ξ ≤ x|B (2.102)


Example 2.74: Let ξ and η be fuzzy variables. Then the conditional credi-bility distribution of ξ given η = y is

Φ(x|η = y) =

Crξ ≤ x, η = yCrη = y

, ifCrξ ≤ x, η = y

Crη = y< 0.5

1− Crξ > x, η = yCrη = y

, ifCrξ > x, η = y

Crη = y< 0.5

0.5, otherwise

provided that Crη = y > 0.

Definition 2.35 (Liu [132]) The conditional credibility density function φof a fuzzy variable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ <, (2.103)

∫ +∞

−∞φ(y|B)dy = 1 (2.104)

where Φ(x|B) is the conditional credibility distribution of ξ given B.

Definition 2.36 (Liu [132]) Let ξ be a fuzzy variable. Then the conditionalexpected value of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Crξ ≥ r|Bdr −∫ 0

−∞Crξ ≤ r|Bdr (2.105)


Following conditional credibility and conditional expected value, we alsohave conditional variance, conditional moments, conditional critical values,conditional entropy as well as conditional convergence.

Definition 2.37 Let ξ be a nonnegative fuzzy variable representing lifetime.Then the hazard rate (or failure rate) is

h(x) = lim∆↓0

Crξ ≤ x+ ∆∣∣ ξ > x. (2.106)

Section 2.16 - Fuzzy Process 119

The hazard rate tells us the credibility of a failure just after time x when itis functioning at time x.

Example 2.75: Let ξ be an exponentially distributed fuzzy variable. Thenits hazard rate h(x) ≡ 0.5. In fact, the hazard rate is always 0.5 if themembership function is positive and decreasing.

Example 2.76: Let ξ be a triangular fuzzy variable (a, b, c) with a ≥ 0.Then its hazard rate

h(x) =

0, if x ≤ a

x− a

b− a, if a ≤ x ≤ (a+ b)/2

0.5, if (a+ b)/2 ≤ x < c

0, if x ≥ c.

2.16 Fuzzy Process

Definition 2.38 Let T be an index set and let (Θ,P,Cr) be a credibilityspace. A fuzzy process is a function from T × (Θ,P,Cr) to the set of realnumbers.

That is, a fuzzy process Xt(θ) is a function of two variables such that thefunction Xt∗(θ) is a fuzzy variable for each t∗. For each fixed θ∗, the functionXt(θ∗) is called a sample path of the fuzzy process. A fuzzy process Xt(θ) issaid to be sample-continuous if the sample path is continuous for almost allθ.

Definition 2.39 A fuzzy process Xt is said to have independent incrementsif

Xt1 −Xt0 , Xt2 −Xt1 , · · · , Xtk−Xtk−1 (2.107)

are independent fuzzy variables for any times t0 < t1 < · · · < tk. A fuzzyprocess Xt is said to have stationary increments if, for any given t > 0, theXs+t −Xs are identically distributed fuzzy variables for all s > 0.

Fuzzy Renewal Process

Definition 2.40 (Zhao and Liu [251]) Let ξ1, ξ2, · · · be iid positive fuzzyvariables. Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn for n ≥ 1. Then thefuzzy process

Nt = maxn≥0

n∣∣ Sn ≤ t

(2.108)

is called a fuzzy renewal process.


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4 ...............

3 ...............

2 ...............

1 ...............

0 ...............

ξ1 ξ2 ξ3 ξ4

S0 S1 S2 S3 S4

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t

Nt

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Figure 2.7: A Sample Path of Fuzzy Renewal Process



CrNt ≥ n = CrSn ≤ t = Crξ1 ≤

t

n

. (2.109)

Theorem 2.65 Let Nt be a fuzzy renewal process with interarrival timesξ1, ξ2, · · · Then we have

E[Nt] =∞∑

n=1

CrSn ≤ t =∞∑

n=1

Crξ1 ≤

t

n

. (2.110)


E[Nt] =∫ ∞

0

CrNt ≥ rdr =∞∑

n=1

∫ n

n−1

CrNt ≥ rdr

=∞∑

n=1

CrNt ≥ n =∞∑

n=1

CrSn ≤ t =∞∑

n=1

Crξ1 ≤

t

n

.


Example 2.77: A renewal process Nt is called an equipossible renewal pro-cess if ξ1, ξ2, · · · are iid equipossible fuzzy variables (a, b) with a > 0. Thenfor each nonnegative integer n, we have

CrNt ≥ n =

0, if t < na

0.5, if na ≤ t < nb

1, if t ≥ nb,

(2.111)


E[Nt] =12

(⌊t

a

⌋+⌊t

b

⌋)(2.112)

where bxc represents the maximal integer less than or equal to x.

Example 2.78: A renewal process Nt is called a triangular renewal processif ξ1, ξ2, · · · are iid triangular fuzzy variables (a, b, c) with a > 0. Then foreach nonnegative integer n, we have

CrNt ≥ n =

0, if t ≤ na

t− na

2n(b− a), if na ≤ t ≤ nb

nc− 2nb+ t

2n(c− b), if nb ≤ t ≤ nc

1, if t ≥ nc.

(2.113)

Theorem 2.66 (Zhao and Liu [251], Renewal Theorem) Let Nt be a fuzzyrenewal process with interarrival times ξ1, ξ2, · · · Then

limt→∞

E[Nt]t

= E

[1ξ1

]. (2.114)

Proof: Since ξ1 is a positive fuzzy variable and Nt is a nonnegative fuzzyvariable, we have

E[Nt]t

=∫ ∞

0

CrNt

t≥ r

dr, E

[1ξ1

]=∫ ∞

0

Cr

1ξ1≥ r

dr.

It is easy to verify that

CrNt

t≥ r

≤ Cr

1ξ1≥ r

and

limt→∞

CrNt

t≥ r

= Cr

1ξ1≥ r

for any real numbers t > 0 and r > 0. It follows from Lebesgue dominatedconvergence theorem that

limt→∞

∫ ∞

0

CrNt

t≥ r

dr =

∫ ∞

0

Cr

1ξ1≥ r

dr.

Hence (2.114) holds.


C Process

This subsection introduces a fuzzy counterpart of Brownian motion, andprovides some basic mathematical properties.

Definition 2.41 (Liu [133]) A fuzzy process Ct is said to be a C process if(i) C0 = 0,(ii) Ct has stationary and independent increments,(iii) every increment Cs+t − Cs is a normally distributed fuzzy variable withexpected value et and variance σ2t2.

The parameters e and σ are called the drift and diffusion coefficients,respectively. The C process is said to be standard if e = 0 and σ = 1. Any Cprocess may be represented by et+ σCt where Ct is a standard C process.

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Figure 2.8: A Sample Path of Standard C Process

Perhaps the readers would like to know why the increment is a normallydistributed fuzzy variable. The reason is that a normally distributed fuzzyvariable has maximum entropy when its expected value and variance aregiven, just like a normally distributed random variable.

Theorem 2.67 (Liu [133], Existence Theorem) There is a C process. Fur-thermore, each version of C process is sample-continuous.

Proof: Without loss of generality, we only prove that there is a standard Cprocess on the range of t ∈ [0, 1]. Let

ξ(r)∣∣ r represents rational numbers in [0, 1]

be a countable sequence of independently and normally distributed fuzzyvariables with expected value zero and variance one. For each integer n, wedefine a fuzzy process

Xn(t) =

1n

k∑i=1

ξ

(i

n

), if t =

k

n(k = 0, 1, · · · , n)

linear, otherwise.


Since the limitlim

n→∞Xn(t)

exists almost surely, we may verify that the limit meets the conditions of Cprocess. Hence there is a standard C process.

Remark 2.9: Suppose that Ct is a standard C process. It has been provedthat

X1(t) = −Ct, (2.115)

X2(t) = aCt/a, (2.116)

X3(t) = Ct+s − Cs (2.117)

are each a version of standard C process.Dai [23] proved that almost all C paths are Lipschitz continuous and

have a finite variation. Thus almost all C paths are differentiable almosteverywhere and have zero squared variation. However, for any number ` ∈(0, 0.5), there is a sample θ with Crθ = ` such that Ct(θ) is not differentiablein a dense set of t.

As a byproduct, C path is an example of Lipschitz continuous functionwhose nondifferentiable points are dense, while Brownian path is an exampleof continuous function that is nowhere differentiable. In other words, C pathis the worst Lipschitz continuous function and Brownian path is the worstcontinuous function in the sense of differentiability.

Example 2.79: Let Ct be a C process with drift 0. Then for any level x > 0and any time t > 0, Dai [23] proved the following reflection principle,

Cr

max0≤s≤t

Cs ≥ x

= CrCt ≥ x. (2.118)

In addition, for any level x < 0 and any time t > 0, we have

Cr

min0≤s≤t

Cs ≤ x

= CrCt ≤ x. (2.119)

Example 2.80: Let Ct be a C process with drift e > 0 and diffusion coeffi-cient σ. Then the first passage time τ that the C process reaches the barrierx > 0 has the membership function

µ(t) = 2(

1 + exp(π|x− et|√

6σt

))−1

, t > 0 (2.120)

whose expected value E[τ ] = +∞ (Dai [23]).


Definition 2.42 (Liu [133]) Let Ct be a standard C process. Then et+ σCt

is a C process, and the fuzzy process

Gt = exp(et+ σCt) (2.121)

is called a geometric C process.

Geometric C process Gt is employed to model stock prices. For each t > 0,Li [108] proved that Gt has a lognormal membership function

µ(z) = 2(

1 + exp(π| ln z − et|√

6σt

))−1

, z ≥ 0 (2.122)

whose expected value is

E[Gt] = exp(et) csc(√

6σt)√

6σt, ∀t < π/(√

6σ). (2.123)

In addition, the first passage time that a geometric C process Gt reaches thebarrier x > 1 is just the time that the C process with drift e and diffusion σreaches lnx.

A Basic Stock Model

It was assumed that stock price follows geometric Brownian motion, andstochastic financial mathematics was then founded based on this assump-tion. Liu [133] presented an alternative assumption that stock price followsgeometric C process. Based on this assumption, we obtain a basic stockmodel for fuzzy financial market in which the bond price Xt and the stockprice Yt follow

Xt = X0 exp(rt)

Yt = Y0 exp(et+ σCt)(2.124)

where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion,and Ct is a standard C process. It is just a fuzzy counterpart of Black-Scholesstock model [8]. For exploring further development of fuzzy stock models,the interested readers may consult Gao [50], Peng [191], Qin and Li [194],and Zhu [267].

2.17 Fuzzy Calculus

Let Ct be a standard C process, and dt an infinitesimal time interval. Then

dCt = Ct+dt − Ct

is a fuzzy process such that, for each t, the dCt is a normally distributedfuzzy variable with

E[dCt] = 0, V [dCt] = dt2,

E[dC2t ] = dt2, V [dC2

t ] ≈ 7dt4.

Section 2.17 - Fuzzy Calculus 125

Definition 2.43 (Liu [133]) Let Xt be a fuzzy process and let Ct be a stan-dard C process. For any partition of closed interval [a, b] with a = t1 < t2 <· · · < tk+1 = b, the mesh is written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the fuzzy integral of Xt with respect to Ct is∫ b

a

XtdCt = lim∆→0

k∑i=1

Xti· (Cti+1 − Cti

) (2.125)

provided that the limit exists almost surely and is a fuzzy variable.

Example 2.81: Let Ct be a standard C process. Then for any partition0 = t1 < t2 < · · · < tk+1 = s, we have∫ s

0

dCt = lim∆→0

k∑i=1

(Cti+1 − Cti) ≡ Cs − C0 = Cs.

Example 2.82: Let Ct be a standard C process. Then for any partition0 = t1 < t2 < · · · < tk+1 = s, we have

sCs =k∑

i=1

(ti+1Cti+1 − tiCti

)=

k∑i=1

ti(Cti+1 − Cti) +

k∑i=1

Cti+1(ti+1 − ti)

→∫ s

0

tdCt +∫ s

0

Ctdt


0

tdCt = sCs −∫ s

0

Ctdt.

Example 2.83: Let Ct be a standard C process. Then for any partition0 = t1 < t2 < · · · < tk+1 = s, we have

C2s =

k∑i=1

(C2

ti+1− C2

ti

)

=k∑

i=1

(Cti+1 − Cti

)2 + 2k∑

i=1

Cti

(Cti+1 − Cti

)→ 0 + 2

∫ s

0

CtdCt



0

CtdCt =12C2

s .

This equation shows that fuzzy integral does not behave like Ito integral. Infact, the fuzzy integral behaves like ordinary integrals.

Theorem 2.68 (Liu [133]) Let Ct be a standard C process, and let h(t, c) bea continuously differentiable function. Define Xt = h(t, Ct). Then we havethe following chain rule

dXt =∂h

∂t(t, Ct)dt+

∂h

∂c(t, Ct)dCt. (2.126)

Proof: Since the function h is continuously differentiable, by using Taylorseries expansion, the infinitesimal increment of Xt has a first-order approxi-mation

∆Xt =∂h

∂t(t, Ct)∆t+

∂h

∂c(t, Ct)∆Ct.

Hence we obtain the chain rule because it makes

Xs = X0 +∫ s

0

∂h

∂t(t, Ct)dt+

∫ s

0

∂h

∂c(t, Ct)dCt

for any s ≥ 0.

Remark 2.10: The infinitesimal increment dCt in (2.126) may be replacedwith the derived C process

dYt = utdt+ vtdCt (2.127)

where ut and vt are absolutely integrable fuzzy processes, thus producing

dh(t, Yt) =∂h

∂t(t, Yt)dt+

∂h

∂c(t, Yt)dYt. (2.128)

Remark 2.11: Assume that C1t, C2t, · · · , Cmt are standard C processes,and h(t, c1, c2, · · · , cm) is a continuously differentiable function. Define

Xt = h(t, C1t, C2t, · · · , Cmt).

You [243] proved the following chain rule

dXt =∂h

∂tdt+

m∑i=1

∂h

∂cidCit. (2.129)

Example 2.84: Applying the chain rule, we obtain the following formula

d(tCt) = Ctdt+ tdCt.

Section 2.18 - Fuzzy Differential Equation 127

Hence we have

sCs =∫ s

0

d(tCt) =∫ s

0

Ctdt+∫ s

0

tdCt.

That is, ∫ s

0

tdCt = sCs −∫ s

0

Ctdt.

Example 2.85: Let Ct be a standard C process. By using the chain rule

d(C2t ) = 2CtdCt,

we get

C2s =

∫ s

0

d(C2t ) = 2

∫ s

0

CtdCt.

It follows that ∫ s

0

CtdCt =12C2

s .

Example 2.86: Let Ct be a standard C process. Then we have the followingchain rule

d(C3t ) = 3C2

t dCt.

Thus we obtain

C3s =

∫ s

0

d(C3t ) = 3

∫ s

0

C2t dCt.

That is ∫ s

0

C2t dCt =

13C3

s .

Theorem 2.69 (Liu [133], Integration by Parts) Suppose that Ct is a stan-dard C process and F (t) is an absolutely continuous function. Then∫ s

0

F (t)dCt = F (s)Cs −∫ s

0

CtdF (t). (2.130)

Proof: By defining h(t, Ct) = F (t)Ct and using the chain rule, we get

d(F (t)Ct) = CtdF (t) + F (t)dCt.

Thus

F (s)Cs =∫ s

0

d(F (t)Ct) =∫ s

0

CtdF (t) +∫ s

0

F (t)dCt



2.18 Fuzzy Differential Equation

Fuzzy differential equation is a type of differential equation driven by C pro-cess.

Definition 2.44 (Liu [133]) Suppose Ct is a standard C process, and f andg are some given functions. Then

dXt = f(t,Xt)dt+ g(t,Xt)dCt (2.131)

is called a fuzzy differential equation. A solution is a fuzzy process Xt thatsatisfies (2.131) identically in t.

Remark 2.12: Note that there is no precise definition for the terms dXt,dt and dCt in the fuzzy differential equation (2.131). The mathematicallymeaningful form is the fuzzy integral equation

Xs = X0 +∫ s

0

f(t,Xt)dt+∫ s

0

g(t,Xt)dCt. (2.132)

However, the differential form is more convenient for us. This is the mainreason why we accept the differential form.

Example 2.87: Let Ct be a standard C process. Then the fuzzy differentialequation

dXt = adt+ bdCt

has a solutionXt = at+ bCt

which is just a C process with drift coefficient a and diffusion coefficient b.

Example 2.88: Let Ct be a standard C process. Then the fuzzy differentialequation

dXt = aXtdt+ bXtdCt

has a solutionXt = exp (at+ bCt)

which is just a geometric C process.

Example 2.89: Let Ct be a standard C process. Then the fuzzy differentialequations

dXt = −YtdCt

dYt = XtdCt

have a solution(Xt, Yt) = (cosCt, sinCt)

which is called a C process on unit circle since X2t + Y 2

t ≡ 1.

Chapter 3

Chance Theory

Fuzziness and randomness are two basic types of uncertainty. In many cases,fuzziness and randomness simultaneously appear in a system. In order todescribe this phenomena, a fuzzy random variable was introduced by Kwak-ernaak [87][88] as a random element taking “fuzzy variable” values. In addi-tion, a random fuzzy variable was proposed by Liu [124] as a fuzzy elementtaking “random variable” values. For example, it might be known that thelifetime of a modern engine is an exponentially distributed random variablewith an unknown parameter. If the parameter is provided as a fuzzy variable,then the lifetime is a random fuzzy variable.

More generally, a hybrid variable was introduced by Liu [130] as a toolto describe the quantities with fuzziness and randomness. Fuzzy randomvariable and random fuzzy variable are instances of hybrid variable. In orderto measure hybrid events, a concept of chance measure was introduced by Liand Liu [103]. Chance theory is a hybrid of probability theory and credibilitytheory. Perhaps the reader would like to know what axioms we should assumefor chance theory. In fact, chance theory will be based on the three axiomsof probability and four axioms of credibility.

The emphasis in this chapter is mainly on chance space, hybrid variable,chance measure, chance distribution, independence, identical distribution,expected value, variance, moments, critical values, entropy, distance, conver-gence almost surely, convergence in chance, convergence in mean, convergencein distribution, conditional chance, hybrid process, hybrid calculus, and hy-brid differential equation.

3.1 Chance Space

Chance theory begins with the concept of chance space that inherits themathematical foundations of both probability theory and credibility theory.

130 Chapter 3 - Chance Theory

Definition 3.1 (Liu [130]) Suppose that (Θ,P,Cr) is a credibility space and(Ω,A,Pr) is a probability space. The product (Θ,P,Cr)× (Ω,A,Pr) is calleda chance space.

The universal set Θ×Ω is clearly the set of all ordered pairs of the form(θ, ω), where θ ∈ Θ and ω ∈ Ω. What is the product σ-algebra P×A? Whatis the product measure Cr× Pr? Let us discuss these two basic problems.

What is the product σ-algebra P×A?

Generally speaking, it is not true that all subsets of Θ × Ω are measurable.Let Λ be a subset of Θ× Ω. Write

Λ(θ) =ω ∈ Ω

∣∣ (θ, ω) ∈ Λ. (3.1)

It is clear that Λ(θ) is a subset of Ω. If Λ(θ) ∈ A holds for each θ ∈ Θ, thenΛ may be regarded as a measurable set.

Definition 3.2 (Liu [132]) Let (Θ,P,Cr)× (Ω,A,Pr) be a chance space. Asubset Λ ⊂ Θ× Ω is called an event if Λ(θ) ∈ A for each θ ∈ Θ.

Example 3.1: Empty set ∅ and universal set Θ× Ω are clearly events.

Example 3.2: Let X ∈ P and Y ∈ A. Then X × Y is a subset of Θ × Ω.Since the set

(X × Y )(θ) =

Y, if θ ∈ X∅, if θ ∈ Xc

is in the σ-algebra A for each θ ∈ Θ, the rectangle X × Y is an event.

Theorem 3.1 (Liu [132]) Let (Θ,P,Cr)× (Ω,A,Pr) be a chance space. Theclass of all events is a σ-algebra over Θ× Ω, and denoted by P×A.

Proof: At first, it is obvious that Θ×Ω ∈ P×A. For any event Λ, we alwayshave

Λ(θ) ∈ A, ∀θ ∈ Θ.

Thus for each θ ∈ Θ, the set

Λc(θ) =ω ∈ Ω

∣∣ (θ, ω) ∈ Λc

= (Λ(θ))c ∈ A

which implies that Λc ∈ P × A. Finally, let Λ1,Λ2, · · · be events. Then foreach θ ∈ Θ, we have( ∞⋃

i=1

Λi

)(θ) =

ω ∈ Ω

∣∣ (θ, ω) ∈∞⋃

i=1

Λi

=

∞⋃i=1

ω ∈ Ω

∣∣ (θ, ω) ∈ Λi

∈ A.

That is, the countable union ∪iΛi ∈ P×A. Hence P×A is a σ-algebra.

Section 3.1 - Chance Space 131

Example 3.3: When Θ× Ω is countable, usually P×A is the power set.

Example 3.4: When Θ×Ω is uncountable, for example Θ×Ω = <2, usuallyP×A is a σ-algebra between the Borel algebra and power set of <2. Let Xbe a nonempty Borel set and let Y be a non-Borel set of real numbers. Itfollows from X × Y 6∈ P×A that P×A is smaller than the power set. It isalso clear that Y ×X ∈ P×A but Y ×X is not a Borel set. Hence P×A islarger than the Borel algebra.

What is the product measure Cr× Pr?

Product probability is a probability measure, and product credibility is acredibility measure. What is the product measure Cr × Pr? We will call itchance measure and define it as follows.

Definition 3.3 (Li and Liu [103]) Let (Θ,P,Cr) × (Ω,A,Pr) be a chancespace. Then a chance measure of an event Λ is defined as

ChΛ =

supθ∈Θ

(Crθ ∧ PrΛ(θ)),

if supθ∈Θ

(Crθ ∧ PrΛ(θ)) < 0.5

1− supθ∈Θ

(Crθ ∧ PrΛc(θ)),

if supθ∈Θ

(Crθ ∧ PrΛ(θ)) ≥ 0.5.

(3.2)

Example 3.5: Take a credibility space (Θ,P,Cr) to be θ1, θ2 with Crθ1 =0.6 and Crθ2 = 0.4, and take a probability space (Ω,A,Pr) to be ω1, ω2with Prω1 = 0.7 and Prω2 = 0.3. Then

Ch(θ1, ω1) = 0.6, Ch(θ2, ω2) = 0.3.

Example 3.6: Take a credibility space (Θ,P,Cr) to be [0, 1] with Crθ ≡1/2, and take a probability space (Ω,A,Pr) to be [0, 1] with Borel algebraand Lebesgue measure. Then for any real numbers a, b ∈ [0, 1], we have

Ch[0, a]× [0, b] =

b, if b < 0.5

0.5, if 0.5 ≤ b < 11, if a = b = 1.

Theorem 3.2 Let (Θ,P,Cr)×(Ω,A,Pr) be a chance space and Ch a chancemeasure. Then we have

Ch∅ = 0, (3.3)

ChΘ× Ω = 1, (3.4)

0 ≤ ChΛ ≤ 1 (3.5)

for any event Λ.


Proof: It follows from the definition immediately.

Theorem 3.3 Let (Θ,P,Cr)×(Ω,A,Pr) be a chance space and Ch a chancemeasure. Then for any event Λ, we have

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ∨ supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≥ 0.5, (3.6)

supθ∈Θ

(Crθ ∧ PrΛ(θ)) + supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≤ 1, (3.7)

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ≤ ChΛ ≤ 1− supθ∈Θ

(Crθ ∧ PrΛc(θ)). (3.8)

Proof: It follows from the basic properties of probability and credibility that

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ∨ supθ∈Θ

(Crθ ∧ PrΛc(θ))

≥ supθ∈Θ

(Crθ ∧ (PrΛ(θ) ∨ PrΛc(θ)))

≥ supθ∈Θ

Crθ ∧ 0.5 = 0.5

andsupθ∈Θ

(Crθ ∧ PrΛ(θ)) + supθ∈Θ

(Crθ ∧ PrΛc(θ))

= supθ1,θ2∈Θ

(Crθ1 ∧ PrΛ(θ1)+ Crθ2 ∧ PrΛc(θ2))

≤ supθ1 6=θ2

(Crθ1+ Crθ2) ∨ supθ∈Θ

(PrΛ(θ)+ PrΛc(θ))

≤ 1 ∨ 1 = 1.

The inequalities (3.8) follows immediately from the above inequalities andthe definition of chance measure.

Theorem 3.4 (Li and Liu [103]) The chance measure is increasing. Thatis,

ChΛ1 ≤ ChΛ2 (3.9)

for any events Λ1 and Λ2 with Λ1 ⊂ Λ2.

Proof: Since Λ1(θ) ⊂ Λ2(θ) and Λc2(θ) ⊂ Λc

1(θ) for each θ ∈ Θ, we have

supθ∈Θ

(Crθ ∧ PrΛ1(θ)) ≤ supθ∈Θ

(Crθ ∧ PrΛ2(θ)),

supθ∈Θ

(Crθ ∧ PrΛc2(θ)) ≤ sup

θ∈Θ(Crθ ∧ PrΛc

1(θ)).

The argument breaks down into three cases.Case 1: sup

θ∈Θ(Crθ ∧ PrΛ2(θ)) < 0.5. For this case, we have

supθ∈Θ

(Crθ ∧ PrΛ1(θ)) < 0.5,


ChΛ2 = supθ∈Θ

(Crθ ∧ PrΛ2(θ)) ≥ supθ∈Θ

(Crθ ∧ PrΛ1(θ) = ChΛ1.

Case 2: supθ∈Θ

(Crθ∧PrΛ2(θ)) ≥ 0.5 and supθ∈Θ

(Crθ∧PrΛ1(θ)) < 0.5.

It follows from Theorem 3.3 that

ChΛ2 ≥ supθ∈Θ

(Crθ ∧ PrΛ2(θ)) ≥ 0.5 > ChΛ1.

Case 3: supθ∈Θ

(Crθ∧PrΛ2(θ)) ≥ 0.5 and supθ∈Θ

(Crθ∧PrΛ1(θ)) ≥ 0.5.

For this case, we have

ChΛ2 = 1−supθ∈Θ

(Crθ∧PrΛc2(θ)) ≥ 1−sup

θ∈Θ(Crθ∧PrΛc

1(θ)) = ChΛ1.

Thus Ch is an increasing measure.

Theorem 3.5 (Li and Liu [103]) The chance measure is self-dual. That is,

ChΛ+ ChΛc = 1 (3.10)

for any event Λ.

Proof: For any event Λ, please note that

ChΛc =

supθ∈Θ

(Crθ ∧ PrΛc(θ)), if supθ∈Θ

(Crθ ∧ PrΛc(θ)) < 0.5

1− supθ∈Θ

(Crθ ∧ PrΛ(θ)), if supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≥ 0.5.

The argument breaks down into three cases.Case 1: sup

θ∈Θ(Crθ ∧ PrΛ(θ)) < 0.5. For this case, we have

supθ∈Θ

(Crθ ∧ PrΛc(θ)) ≥ 0.5,

ChΛ+ChΛc = supθ∈Θ

(Crθ∧PrΛ(θ))+1− supθ∈Θ

(Crθ∧PrΛ(θ)) = 1.

Case 2: supθ∈Θ

(Crθ ∧PrΛ(θ)) ≥ 0.5 and supθ∈Θ

(Crθ ∧PrΛc(θ)) < 0.5.

For this case, we have

ChΛ+ChΛc = 1−supθ∈Θ

(Crθ∧PrΛc(θ))+supθ∈Θ

(Crθ∧PrΛc(θ)) = 1.

Case 3: supθ∈Θ

(Crθ ∧PrΛ(θ)) ≥ 0.5 and supθ∈Θ

(Crθ ∧PrΛc(θ)) ≥ 0.5.

For this case, it follows from Theorem 3.3 that

supθ∈Θ

(Crθ ∧ PrΛ(θ)) = supθ∈Θ

(Crθ ∧ PrΛc(θ)) = 0.5.

Hence ChΛ+ ChΛc = 0.5 + 0.5 = 1. The theorem is proved.


Theorem 3.6 (Li and Liu [103]) For any event X × Y , we have

ChX × Y = CrX ∧ PrY . (3.11)

Proof: The argument breaks down into three cases.Case 1: CrX < 0.5. For this case, we have

supθ∈X

Crθ ∧ PrY = CrX ∧ CrY < 0.5,

ChX × Y = supθ∈X

Crθ ∧ PrY = CrX ∧ PrY .

Case 2: CrX ≥ 0.5 and PrY < 0.5. Then we have

supθ∈X

Crθ ≥ 0.5,

supθ∈X

Crθ ∧ PrY = PrY < 0.5,

ChX × Y = supθ∈X

Crθ ∧ PrY = PrY = CrX ∧ PrY .

Case 3: CrX ≥ 0.5 and PrY ≥ 0.5. Then we have

supθ∈Θ

(Crθ ∧ Pr(X × Y )(θ)) ≥ supθ∈X

Crθ ∧ PrY ≥ 0.5,

ChX × Y = 1− supθ∈Θ

(Crθ ∧ Pr(X × Y )c(θ)) = CrX ∧ PrY .


Example 3.7: It follows from Theorem 3.6 that for any events X × Ω andΘ× Y , we have

ChX × Ω = CrX, ChΘ× Y = PrY . (3.12)

Theorem 3.7 (Li and Liu [103], Chance Subadditivity Theorem) The chancemeasure is subadditive. That is,

ChΛ1 ∪ Λ2 ≤ ChΛ1+ ChΛ2 (3.13)

for any events Λ1 and Λ2. In fact, chance measure is not only finitely sub-additive but also countably subadditive.

Proof: The proof breaks down into three cases.Case 1: ChΛ1 ∪ Λ2 < 0.5. Then ChΛ1 < 0.5, ChΛ2 < 0.5 and

ChΛ1 ∪ Λ2 = supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)(θ))

≤ supθ∈Θ

(Crθ ∧ (PrΛ1(θ)+ PrΛ2(θ)))

≤ supθ∈Θ

(Crθ ∧ PrΛ1(θ)+ Crθ ∧ PrΛ2(θ))

≤ supθ∈Θ

(Crθ ∧ PrΛ1(θ)) + supθ∈Θ

(Crθ ∧ PrΛ2(θ))

= ChΛ1+ ChΛ2.


Case 2: ChΛ1 ∪ Λ2 ≥ 0.5 and ChΛ1 ∨ ChΛ2 < 0.5. We first have

supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)(θ)) ≥ 0.5.

For any sufficiently small number ε > 0, there exists a point θ such that

Crθ ∧ Pr(Λ1 ∪ Λ2)(θ) > 0.5− ε > ChΛ1 ∨ ChΛ2,

Crθ > 0.5− ε > PrΛ1(θ),

Crθ > 0.5− ε > PrΛ2(θ).

Thus we have

Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Crθ ∧ PrΛ1(θ)+ Crθ ∧ PrΛ2(θ)

= Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ PrΛ1(θ)+ PrΛ2(θ)

≥ Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ) ≥ 1− 2ε

because if Crθ ≥ Pr(Λ1 ∪ Λ2)c(θ), then

Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ)

= Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ)

= 1 ≥ 1− 2ε

and if Crθ < Pr(Λ1 ∪ Λ2)c(θ), then

Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Pr(Λ1 ∪ Λ2)(θ)

= Crθ+ Pr(Λ1 ∪ Λ2)(θ)

≥ (0.5− ε) + (0.5− ε) = 1− 2ε.

Taking supremum on both sides and letting ε→ 0, we obtain

ChΛ1 ∪ Λ2 = 1− supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ))

≤ supθ∈Θ

(Crθ ∧ PrΛ1(θ)) + supθ∈Θ

(Crθ ∧ PrΛ2(θ))

= ChΛ1+ ChΛ2.

Case 3: ChΛ1 ∪ Λ2 ≥ 0.5 and ChΛ1 ∨ ChΛ2 ≥ 0.5. Without lossof generality, suppose ChΛ1 ≥ 0.5. For each θ, we first have

Crθ ∧ PrΛc1(θ) = Crθ ∧ Pr(Λc

1(θ) ∩ Λc2(θ)) ∪ (Λc

1(θ) ∩ Λ2(θ))

≤ Crθ ∧ (Pr(Λ1 ∪ Λ2)c(θ)+ PrΛ2(θ))

≤ Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ)+ Crθ ∧ PrΛ2(θ),


i.e., Crθ∧Pr(Λ1 ∪Λ2)c(θ) ≥ Crθ∧PrΛc1(θ)−Crθ∧PrΛ2(θ). It

follows from Theorem 3.3 that

ChΛ1 ∪ Λ2 = 1− supθ∈Θ

(Crθ ∧ Pr(Λ1 ∪ Λ2)c(θ))

≤ 1− supθ∈Θ

(Crθ ∧ PrΛc1(θ)) + sup

θ∈Θ(Crθ ∧ PrΛ2(θ))

≤ ChΛ1+ ChΛ2.


Remark 3.1: For any events Λ1 and Λ2, it follows from the chance subaddi-tivity theorem that the chance measure is null-additive, i.e., ChΛ1 ∪ Λ2 =ChΛ1+ ChΛ2 if either ChΛ1 = 0 or ChΛ2 = 0.

Theorem 3.8 Let Λi be a decreasing sequence of events with ChΛi → 0as i→∞. Then for any event Λ, we have

limi→∞

ChΛ ∪ Λi = limi→∞

ChΛ\Λi = ChΛ. (3.14)

Proof: Since chance measure is increasing and subadditive, we immediatelyhave

ChΛ ≤ ChΛ ∪ Λi ≤ ChΛ+ ChΛi

for each i. Thus we get ChΛ ∪ Λi → ChΛ by using ChΛi → 0. Since(Λ\Λi) ⊂ Λ ⊂ ((Λ\Λi) ∪ Λi), we have

ChΛ\Λi ≤ ChΛ ≤ ChΛ\Λi+ ChΛi.

Hence ChΛ\Λi → ChΛ by using ChΛi → 0.

Theorem 3.9 (Li and Liu [103], Chance Semicontinuity Law) For eventsΛ1,Λ2, · · · , we have

limi→∞

ChΛi = Ch

limi→∞

Λi

(3.15)

if one of the following conditions is satisfied:(a) ChΛ ≤ 0.5 and Λi ↑ Λ; (b) lim

i→∞ChΛi < 0.5 and Λi ↑ Λ;

(c) ChΛ ≥ 0.5 and Λi ↓ Λ; (d) limi→∞

ChΛi > 0.5 and Λi ↓ Λ.

Proof: (a) Assume ChΛ ≤ 0.5 and Λi ↑ Λ. We first have

ChΛ = supθ∈Θ

(Crθ ∧ PrΛ(θ)), ChΛi = supθ∈Θ

(Crθ ∧ PrΛi(θ))

for i = 1, 2, · · · For each θ ∈ Θ, since Λi(θ) ↑ Λ(θ), it follows from theprobability continuity theorem that

limi→∞

Crθ ∧ PrΛi(θ) = Crθ ∧ PrΛ(θ).

Section 3.2 - Hybrid Variables 137

Taking supremum on both sides, we obtain

limi→∞

supθ∈Θ

(Crθ ∧ PrΛi(θ)) = supθ∈Θ

(Crθ ∧ PrΛ(θ).

The part (a) is verified.(b) Assume limi→∞ ChΛi < 0.5 and Λi ↑ Λ. For each θ ∈ Θ, since

Crθ ∧ PrΛ(θ) = limi→∞

Crθ ∧ PrΛi(θ),

we have

supθ∈Θ

(Crθ ∧ PrΛ(θ)) ≤ limi→∞

supθ∈Θ

(Crθ ∧ PrΛi(θ)) < 0.5.

It follows that ChΛ < 0.5 and the part (b) holds by using (a).(c) Assume ChΛ ≥ 0.5 and Λi ↓ Λ. We have ChΛc ≤ 0.5 and Λc

i ↑ Λc.It follows from (a) that

limi→∞

ChΛi = 1− limi→∞

ChΛci = 1− ChΛc = ChΛ.

(d) Assume limi→∞ ChΛi > 0.5 and Λi ↓ Λ. We have limi→∞

ChΛci <

0.5 and Λci ↑ Λc. It follows from (b) that

limi→∞

ChΛi = 1− limi→∞

ChΛci = 1− ChΛc = ChΛ.


Theorem 3.10 (Chance Asymptotic Theorem) For any events Λ1,Λ2, · · · ,we have

limi→∞

ChΛi ≥ 0.5, if Λi ↑ Θ× Ω, (3.16)

limi→∞

ChΛi ≤ 0.5, if Λi ↓ ∅. (3.17)

Proof: Assume Λi ↑ Θ × Ω. If limi→∞ ChΛi < 0.5, it follows from thechance semicontinuity law that

ChΘ× Ω = limi→∞

ChΛi < 0.5

which is in contradiction with CrΘ×Ω = 1. The first inequality is proved.The second one may be verified similarly.

3.2 Hybrid Variables

Recall that a random variable is a measurable function from a probabilityspace to the set of real numbers, and a fuzzy variable is a function from acredibility space to the set of real numbers. In order to describe a quan-tity with both fuzziness and randomness, we introduce a concept of hybridvariable as follows.


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Credibility Space

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RandomVariable

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•

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Probability Space

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FuzzyVariable

HybridVariable

Figure 3.1: Graphical Representation of Hybrid Variable

Definition 3.4 (Liu [130]) A hybrid variable is a measurable function froma chance space (Θ,P,Cr)× (Ω,A,Pr) to the set of real numbers, i.e., for anyBorel set B of real numbers, the set

ξ ∈ B = (θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∈ B (3.18)

is an event.

Remark 3.2: A hybrid variable degenerates to a fuzzy variable if the valueof ξ(θ, ω) does not vary with ω. For example,

ξ(θ, ω) = θ, ξ(θ, ω) = θ2 + 1, ξ(θ, ω) = sin θ.

Remark 3.3: A hybrid variable degenerates to a random variable if thevalue of ξ(θ, ω) does not vary with θ. For example,

ξ(θ, ω) = ω, ξ(θ, ω) = ω2 + 1, ξ(θ, ω) = sinω.

Remark 3.4: For each fixed θ ∈ Θ, it is clear that the hybrid variable ξ(θ, ω)is a measurable function from the probability space (Ω,A,Pr) to the set ofreal numbers. Thus it is a random variable and we will denote it by ξ(θ, ·).Then a hybrid variable ξ(θ, ω) may also be regarded as a function from acredibility space (Θ,P,Cr) to the set ξ(θ, ·)|θ ∈ Θ of random variables.Thus ξ is a random fuzzy variable defined by Liu [124].

Remark 3.5: For each fixed ω ∈ Ω, it is clear that the hybrid variableξ(θ, ω) is a function from the credibility space (Θ,P,Cr) to the set of real


numbers. Thus it is a fuzzy variable and we will denote it by ξ(·, ω). Then ahybrid variable ξ(θ, ω) may be regarded as a function from a probability space(Ω,A,Pr) to the set ξ(·, ω)|ω ∈ Ω of fuzzy variables. If Crξ(·, ω) ∈ B isa measurable function of ω for any Borel set B of real numbers, then ξ is afuzzy random variable in the sense of Liu and Liu [140].

Model I

If a is a fuzzy variable and η is a random variable, then the sum ξ = a+ η isa hybrid variable. The product ξ = a · η is also a hybrid variable. Generallyspeaking, if f : <2 → < is a measurable function, then

ξ = f(a, η) (3.19)

is a hybrid variable. Suppose that a has a membership function µ, and η hasa probability density function φ. Then for any Borel set B of real numbers,Qin and Liu [195] proved the following formula,

Chf(a, η) ∈ B =

supx

(µ(x)

2∧∫

f(x,y)∈B

φ(y)dy

),

if supx

(µ(x)

2∧∫

f(x,y)∈B

φ(y)dy

)< 0.5

1− supx

(µ(x)

2∧∫

f(x,y)∈Bc

φ(y)dy

),

if supx

(µ(x)

2∧∫

f(x,y)∈B

φ(y)dy

)≥ 0.5.

More generally, let a1, a2, · · · , am be fuzzy variables, and let η1, η2, · · · , ηn berandom variables. If f : <m+n → < is a measurable function, then

ξ = f(a1, a2, · · · , am; η1, η2, · · · , ηn) (3.20)

is a hybrid variable. The chance Chf(a1, a2, · · · , am; η1, η2, · · · , ηn) ∈ Bmay be calculated in a similar way provided that µ is the joint membershipfunction and φ is the joint probability density function.

Model II

Let a1, a2, · · · , am be fuzzy variables, and let p1, p2, · · · , pm be nonnegativenumbers with p1 + p2 + · · ·+ pm = 1. Then

ξ =

a1 with probability p1

a2 with probability p2

· · ·am with probability pm

(3.21)


is clearly a hybrid variable. If a1, a2, · · · , am have membership functionsµ1, µ2, · · · , µm, respectively, then for any Borel set B of real numbers, wehave ([195])

Chξ ∈ B =

supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ B

),

if supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ B

)< 0.5

1− supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ Bc

),

if supx1,x2··· ,xm

((min

1≤i≤m

µi(xi)2

)∧

m∑i=1

pi |xi ∈ B

)≥ 0.5.

Model III

Let η1, η2, · · · , ηm be random variables, and let u1, u2, · · · , um be nonnegativenumbers with u1 ∨ u2 ∨ · · · ∨ um = 1. Then

ξ =

η1 with membership degree u1

η2 with membership degree u2

· · ·ηm with membership degree um

(3.22)

is clearly a hybrid variable. If η1, η2, · · · , ηm have probability density func-tions φ1, φ2, · · · , φm, respectively, then for any Borel set B of real numbers,we have ([195])

Chξ ∈ B =

max1≤i≤m

(ui

2∧∫

B

φi(x)dx),

if max1≤i≤m

(ui

2∧∫

B

φi(x)dx)< 0.5

1− max1≤i≤m

(ui

2∧∫

Bc

φi(x)dx),

if max1≤i≤m

(ui

2∧∫

B

φi(x)dx)≥ 0.5.

Model IV

In many statistics problems, the probability density function is completelyknown except for the values of one or more parameters. For example, it


might be known that the lifetime ξ of a modern engine is an exponentiallydistributed random variable with an unknown expected value β. Usually,there is some relevant information in practice. It is thus possible to specifyan interval in which the value of β is likely to lie, or to give an approximateestimate of the value of β. It is typically not possible to determine the valueof β exactly. If the value of β is provided as a fuzzy variable, then ξ is ahybrid variable. More generally, suppose that ξ has a probability densityfunction

φ(x; a1, a2, · · · , am), x ∈ < (3.23)

in which the parameters a1, a2, · · · , am are fuzzy variables rather than crispnumbers. Then ξ is a hybrid variable provided that φ(x; y1, y2, · · · , ym) isa probability density function for any (y1, y2, · · · , ym) that (a1, a2, · · · , am)may take. If a1, a2, · · · , am have membership functions µ1, µ2, · · · , µm, re-spectively, then for any Borel set B of real numbers, the chance Chξ ∈ Bis ([195])

supy1,y2··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

B

φ(x; y1, y2, · · · , ym)dx),

if supy1,y2,··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

B

φ(x; y1, y2, · · · , ym)dx)< 0.5

1− supy1,y2··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

Bc

φ(x; y1, y2, · · · , ym)dx),

if supy1,y2,··· ,ym

((min

1≤i≤m

µi(yi)2

)∧∫

B

φ(x; y1, y2, · · · , ym)dx)≥ 0.5.

Model V

Suppose a fuzzy variable ξ has a normal membership function with unknownexpected value e and variance σ. If e and σ are provided as random variables,then ξ is a hybrid variable. More generally, suppose that ξ has a membershipfunction

µ(x; η1, η2, · · · , ηm), x ∈ < (3.24)

in which the parameters η1, η2, · · · , ηm are random variables rather than de-terministic numbers. Then ξ is a hybrid variable if µ(x; y1, y2, · · · , ym) is amembership function for any (y1, y2, · · · , ym) that (η1, η2, · · · , ηm) may take.

When are two hybrid variables equal to each other?

Definition 3.5 Let ξ1 and ξ2 be hybrid variables defined on the chance space(Θ,P,Cr) × (Ω,A,Pr). We say ξ1 = ξ2 if ξ1(θ, ω) = ξ2(θ, ω) for almost all(θ, ω) ∈ Θ× Ω.


Hybrid Vectors

Definition 3.6 An n-dimensional hybrid vector is a measurable functionfrom a chance space (Θ,P,Cr) × (Ω,A,Pr) to the set of n-dimensional realvectors, i.e., for any Borel set B of <n, the set

ξ ∈ B =(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ∈ B

(3.25)

is an event.

Theorem 3.11 The vector (ξ1, ξ2, · · · , ξn) is a hybrid vector if and only ifξ1, ξ2, · · · , ξn are hybrid variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is a hybrid vector onthe chance space (Θ,P,Cr) × (Ω,A,Pr). For any Borel set B of <, the setB ×<n−1 is a Borel set of <n. Thus the set

(θ, ω) ∈ Θ× Ω∣∣ ξ1(θ, ω) ∈ B

=(θ, ω) ∈ Θ× Ω

∣∣ ξ1(θ, ω) ∈ B, ξ2(θ, ω) ∈ <, · · · , ξn(θ, ω) ∈ <

=(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ∈ B ×<n−1

is an event. Hence ξ1 is a hybrid variable. A similar process may prove thatξ2, ξ3, · · · , ξn are hybrid variables.

Conversely, suppose that all ξ1, ξ2, · · · , ξn are hybrid variables on thechance space (Θ,P,Cr)× (Ω,A,Pr). We define

B =B ⊂ <n

∣∣ (θ, ω) ∈ Θ× Ω|ξ(θ, ω) ∈ B is an event.

The vector ξ = (ξ1, ξ2, · · · , ξn) is proved to be a hybrid vector if we canprove that B contains all Borel sets of <n. First, the class B contains allopen intervals of <n because

(θ, ω)∣∣ ξ(θ, ω) ∈

n∏i=1

(ai, bi)

=

n⋂i=1

(θ, ω)

∣∣ ξi(θ, ω) ∈ (ai, bi)

is an event. Next, the class B is a σ-algebra of <n because (i) we have <n ∈ Bsince (θ, ω)|ξ(θ, ω) ∈ <n = Θ× Ω; (ii) if B ∈ B, then

(θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∈ B

is an event, and

(θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∈ Bc = (θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ∈ Bc

is an event. This means that Bc ∈ B; (iii) if Bi ∈ B for i = 1, 2, · · · , then(θ, ω) ∈ Θ× Ω|ξ(θ, ω) ∈ Bi are events and

(θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∈

∞⋃i=1

Bi

=

∞⋃i=1

(θ, ω) ∈ Θ× Ω∣∣ ξ(θ, ω) ∈ Bi

is an event. This means that ∪iBi ∈ B. Since the smallest σ-algebra con-taining all open intervals of <n is just the Borel algebra of <n, the class Bcontains all Borel sets of <n. The theorem is proved.

Section 3.3 - Chance Distribution 143

Hybrid Arithmetic

Definition 3.7 Let f : <n → < be a measurable function, and ξ1, ξ2, · · · , ξnhybrid variables on the chance space (Θ,P,Cr) × (Ω,A,Pr). Then ξ =f(ξ1, ξ2, · · · , ξn) is a hybrid variable defined as

ξ(θ, ω) = f(ξ1(θ, ω), ξ2(θ, ω), · · · , ξn(θ, ω)), ∀(θ, ω) ∈ Θ× Ω. (3.26)

Example 3.8: Let ξ1 and ξ2 be two hybrid variables. Then the sum ξ =ξ1 + ξ2 is a hybrid variable defined by

ξ(θ, ω) = ξ1(θ, ω) + ξ2(θ, ω), ∀(θ, ω) ∈ Θ× Ω.

The product ξ = ξ1ξ2 is also a hybrid variable defined by

ξ(θ, ω) = ξ1(θ, ω) · ξ2(θ, ω), ∀(θ, ω) ∈ Θ× Ω.

Theorem 3.12 Let ξ be an n-dimensional hybrid vector, and f : <n → < ameasurable function. Then f(ξ) is a hybrid variable.

Proof: Assume that ξ is a hybrid vector on the chance space (Θ,P,Cr) ×(Ω,A,Pr). For any Borel set B of <, since f is a measurable function, thef−1(B) is a Borel set of <n. Thus the set

(θ, ω) ∈ Θ× Ω∣∣ f(ξ(θ, ω)) ∈ B

=(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ∈ f−1(B)

is an event for any Borel set B. Hence f(ξ) is a hybrid variable.

3.3 Chance Distribution

Chance distribution has been defined in several ways. Yang and Liu [240]presented the concept of chance distribution of fuzzy random variables, andZhu and Liu [261] proposed the chance distribution of random fuzzy variables.Li and Liu [103] gave the following definition of chance distribution of hybridvariables.

Definition 3.8 The chance distribution Φ: < → [0, 1] of a hybrid variable ξis defined by

Φ(x) = Ch(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ≤ x. (3.27)

Example 3.9: Let η be a random variable on a probability space (Ω,A,Pr).It is clear that η may be regarded as a hybrid variable on the chance space(Θ,P,Cr)× (Ω,A,Pr) as follows,

ξ(θ, ω) = η(ω), ∀(θ, ω) ∈ Θ× Ω.


Thus its chance distribution is

Φ(x) = Ch(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ≤ x

= ChΘ× ω ∈ Ω

∣∣ η(ω) ≤ x

= CrΘ ∧ Prω ∈ Ω

∣∣ η(ω) ≤ x

= Prη ≤ x

which is just the probability distribution of the random variable η.

Example 3.10: Let a be a fuzzy variable on a credibility space (Θ,P,Cr).It is clear that a may be regarded as a hybrid variable on the chance space(Θ,P,Cr)× (Ω,A,Pr) as follows,

ξ(θ, ω) = a(θ), ∀(θ, ω) ∈ Θ× Ω.

Thus its chance distribution is

Φ(x) = Ch(θ, ω) ∈ Θ× Ω

∣∣ ξ(θ, ω) ≤ x

= Chθ ∈ Θ

∣∣ a(θ) ≤ x × Ω

= Crθ ∈ Θ

∣∣ a(θ) ≤ x∧ PrΩ

= Cra ≤ x

which is just the credibility distribution of the fuzzy variable a.

Theorem 3.13 (Sufficient and Necessary Condition for Chance Distribu-tion) A function Φ : < → [0, 1] is a chance distribution if and only if it is anincreasing function with

limx→−∞

Φ(x) ≤ 0.5 ≤ limx→+∞

Φ(x), (3.28)

limy↓x

Φ(y) = Φ(x) if limy↓x

Φ(y) > 0.5 or Φ(x) ≥ 0.5. (3.29)

Proof: It is obvious that a chance distribution Φ is an increasing function.The inequalities (3.28) follow from the chance asymptotic theorem immedi-ately. Assume that x is a point at which limy↓x Φ(y) > 0.5. That is,

limy↓x

Chξ ≤ y > 0.5.

Since ξ ≤ y ↓ ξ ≤ x as y ↓ x, it follows from the chance semicontinuitylaw that

Φ(y) = Chξ ≤ y ↓ Chξ ≤ x = Φ(x)

as y ↓ x. When x is a point at which Φ(x) ≥ 0.5, if limy↓x Φ(y) 6= Φ(x), thenwe have

limy↓x

Φ(y) > Φ(x) ≥ 0.5.


For this case, we have proved that limy↓x Φ(y) = Φ(x). Thus (3.29) is proved.Conversely, suppose Φ : < → [0, 1] is an increasing function satisfying

(3.28) and (3.29). Theorem 2.19 states that there is a fuzzy variable whosecredibility distribution is just Φ(x). Since a fuzzy variable is a special hybridvariable, the theorem is proved.

Definition 3.9 The chance density function φ: < → [0,+∞) of a hybridvariable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy, ∀x ∈ <, (3.30)

∫ +∞

−∞φ(y)dy = 1 (3.31)

where Φ is the chance distribution of ξ.

Theorem 3.14 Let ξ be a hybrid variable whose chance density function φexists. Then we have

Chξ ≤ x =∫ x

−∞φ(y)dy, Chξ ≥ x =

∫ +∞

x

φ(y)dy. (3.32)

Proof: The first part follows immediately from the definition. In addition,by the self-duality of chance measure, we have

Chξ ≥ x = 1− Chξ < x =∫ +∞

−∞φ(y)dy −

∫ x

−∞φ(y)dy =

∫ +∞

x

φ(y)dy.


Joint Chance Distribution

Definition 3.10 Let (ξ1, ξ2, · · · , ξn) be a hybrid vector. Then the joint chancedistribution Φ : <n → [0, 1] is defined by

Φ(x1, x2, · · · , xn) = Ch ξ1 ≤ x1, ξ2 ≤ x2, · · · , ξn ≤ xn .

Definition 3.11 The joint chance density function φ : <n → [0,+∞) of ahybrid vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn


−∞

∫ +∞

−∞· · ·∫ +∞

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn = 1

where Φ is the joint chance distribution of the hybrid vector (ξ1, ξ2, · · · , ξn).


3.4 Expected Value

Expected value has been defined in several ways. For example, Kwakernaak[87], Puri and Ralescu [193], Kruse and Meyer [86], and Liu and Liu [140]gave different expected value operators of fuzzy random variables. Liu andLiu [141] presented an expected value operator of random fuzzy variables. Liand Liu [103]) suggested the following definition of expected value operatorof hybrid variables.

Definition 3.12 Let ξ be a hybrid variable. Then the expected value of ξ isdefined by

E[ξ] =∫ +∞

0

Chξ ≥ rdr −∫ 0

−∞Chξ ≤ rdr (3.33)


Example 3.11: If a hybrid variable ξ degenerates to a random variable η,then

Chξ ≤ x = Prη ≤ x, Chξ ≥ x = Prη ≥ x, ∀x ∈ <.

It follows from (3.33) that E[ξ] = E[η]. In other words, the expected valueoperator of hybrid variable coincides with that of random variable.

Example 3.12: If a hybrid variable ξ degenerates to a fuzzy variable a, then

Chξ ≤ x = Cra ≤ x, Chξ ≥ x = Cra ≥ x, ∀x ∈ <.

It follows from (3.33) that E[ξ] = E[a]. In other words, the expected valueoperator of hybrid variable coincides with that of fuzzy variable.

Example 3.13: Let a be a fuzzy variable and η a random variable withfinite expected values. Then the hybrid variable ξ = a+η has expected valueE[ξ] = E[a] + E[η].

Theorem 3.15 Let ξ be a hybrid variable whose chance density function φexists. If the Lebesgue integral ∫ +∞

−∞xφ(x)dx


E[ξ] =∫ +∞

−∞xφ(x)dx. (3.34)



E[ξ] =∫ +∞

0

Chξ ≥ rdr −∫ 0

−∞Chξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.


Theorem 3.16 Let ξ be a hybrid variable with chance distribution Φ. If

limx→−∞

Φ(x) = 0, limx→+∞

Φ(x) = 1


−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (3.35)


diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0


∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y


∫ y

−∞xdΦ(x) = 0.


y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y (1− Φ(y)) ≥ 0, for y > 0,


∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, for y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Chξ ≥ rdr


n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1)(xi+1 − xi) = y(Φ(y)− 1) → 0


0

Chξ ≥ rdr =∫ +∞

0

xdΦ(x).


−∫ 0

−∞Chξ ≤ rdr =

∫ 0

−∞xdΦ(x).


Theorem 3.17 Let ξ be a hybrid variable with finite expected values. Thenfor any real numbers a and b, we have

E[aξ + b] = aE[ξ] + b. (3.36)


E[ξ + b] =∫ +∞

0

Chξ + b ≥ rdr −∫ 0

−∞Chξ + b ≤ rdr

=∫ +∞

0

Chξ ≥ r − bdr −∫ 0

−∞Chξ ≤ r − bdr

= E[ξ] +∫ b

0

(Chξ ≥ r − b+ Chξ < r − b)dr

= E[ξ] + b.



E[aξ + b] = E[ξ]−∫ 0

b

(Chξ ≥ r − b+ Chξ < r − b)dr = E[ξ] + b.

Step 2: We prove E[aξ] = aE[ξ]. If a = 0, then the equation E[aξ] =aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ +∞

0

Chaξ ≥ rdr −∫ 0

−∞Chaξ ≤ rdr

=∫ +∞

0

Chξ ≥ r/adr −∫ 0

−∞Chξ ≤ r/adr

= a

∫ +∞

0

Chξ ≥ tdt− a

∫ 0

−∞Chξ ≤ tdt

= aE[ξ].

If a < 0, we have

E[aξ] =∫ +∞

0

Chaξ ≥ rdr −∫ 0

−∞Chaξ ≤ rdr

=∫ +∞

0

Chξ ≤ r/adr −∫ 0

−∞Chξ ≥ r/adr

= a

∫ +∞

0

Chξ ≥ tdt− a

∫ 0

−∞Chξ ≤ tdt

= aE[ξ].

Step 2: For any real numbers a and b, it follows from Steps 1 and 2 that

E[aξ + b] = E[aξ] + b = aE[ξ] + b.


3.5 Variance

The variance has been given by different ways. Liu and Liu [140][141] pro-posed the variance definitions of fuzzy random variables and random fuzzyvariables. Li and Liu [103] suggested the following variance definition ofhybrid variables.

Definition 3.13 Let ξ be a hybrid variable with finite expected value e. Thenthe variance of ξ is defined by V [ξ] = E[(ξ − e)2].

Theorem 3.18 If ξ is a hybrid variable with finite expected value, a and bare real numbers, then V [aξ + b] = a2V [ξ].



V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 3.19 Let ξ be a hybrid variable with expected value e. Then V [ξ] =0 if and only if Chξ = e = 1.


E[(ξ − e)2] =∫ +∞

0

Ch(ξ − e)2 ≥ rdr

which implies Ch(ξ − e)2 ≥ r = 0 for any r > 0. Hence we have

Ch(ξ − e)2 = 0 = 1.

That is, Chξ = e = 1.Conversely, if Chξ = e = 1, then we have Ch(ξ − e)2 = 0 = 1 and

Ch(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

Ch(ξ − e)2 ≥ rdr = 0.



Theorem 3.20 Let f be a convex function on [a, b], and ξ a hybrid variablethat takes values in [a, b] and has expected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (3.37)

Proof: For each (θ, ω) ∈ Θ× Ω, we have a ≤ ξ(θ, ω) ≤ b and

ξ(θ, ω) =b− ξ(θ, ω)b− a

a+ξ(θ, ω)− a

b− ab.


f(ξ(θ, ω)) ≤ b− ξ(θ, ω)b− a

f(a) +ξ(θ, ω)− a

b− af(b).


Theorem 3.21 (Maximum Variance Theorem) Let ξ be a hybrid variablethat takes values in [a, b] and has expected value e. Then

V [ξ] ≤ (e− a)(b− e). (3.38)

Proof: It follows from Theorem 3.20 immediately by defining f(x) = (x−e)2.


3.6 Moments

Liu [129] defined the concepts of moments of both fuzzy random variables andrandom fuzzy variables. Li and Liu [103] discussed the moments of hybridvariables.

Definition 3.14 Let ξ be a hybrid variable. Then for any positive integer k,(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.


Theorem 3.22 Let ξ be a nonnegative hybrid variable, and k a positive num-ber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1Chξ ≥ rdr. (3.39)


E[ξk] =∫ ∞

0

Chξk ≥ xdx =∫ ∞

0

Chξ ≥ rdrk = k

∫ ∞

0

rk−1Chξ ≥ rdr.


Theorem 3.23 (Li and Liu [104]) Let ξ be a hybrid variable that takes val-ues in [a, b] and has expected value e. Then for any positive integer k, thekth absolute moment and kth absolute central moment satisfy the followinginequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (3.40)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (3.41)


3.7 Independence

This book uses the following definition of independence of hybrid variables.

Definition 3.15 (Liu [132]) The hybrid variables ξ1, ξ2, · · · , ξn are said tobe independent if

E

[n∑

i=1

fi(ξi)

]=

n∑i=1

E[fi(ξi)] (3.42)


for any measurable functions f1, f2, · · · , fn provided that the expected valuesexist and are finite.

Theorem 3.24 Hybrid variables are independent if they are (independent ornot) random variables.

Proof: Suppose the hybrid variables are random variables ξ1, ξ2, · · · , ξn.Then f1(ξ1), f2(ξ2), · · · , fn(ξn) are also random variables for any measurablefunctions f1, f2, · · · , fn. It follows from the linearity of expected value oper-ator of random variables that

E[f1(ξ1) + f2(ξ2) + · · ·+ fn(ξn)] = E[f1(ξ1)] + E[f2(ξ2)] + · · ·+ E[fn(ξn)].

Hence the hybrid variables are independent.

Theorem 3.25 Hybrid variables are independent if they are independentfuzzy variables.

Proof: If the hybrid variables are independent fuzzy variables ξ1, ξ2, · · · , ξn,then f1(ξ1), f2(ξ2), · · · , fn(ξn) are also independent fuzzy variables for anymeasurable functions f1, f2, · · · , fn. It follows from the linearity of expectedvalue operator of fuzzy variables that

E[f1(ξ1) + f2(ξ2) + · · ·+ fn(ξn)] = E[f1(ξ1)] + E[f2(ξ2)] + · · ·+ E[fn(ξn)].

Hence the hybrid variables are independent.

Theorem 3.26 Two hybrid variables are independent if one is a randomvariable and another is a fuzzy variable.

Proof: Suppose that ξ is a random variable and η is a fuzzy variable. Thenf(ξ) is a random variable and g(η) is a fuzzy variable for any measurablefunctions f and g. Thus

E[f(ξ) + g(η)] = E[f(ξ)] + E[g(η)].

Hence ξ and η are independent hybrid variables.

Theorem 3.27 If ξ and η are independent hybrid variables with finite ex-pected values, then we have

E[aξ + bη] = aE[ξ] + bE[η] (3.43)

for any real numbers a and b.

Proof: The theorem follows from the definition by defining f1(x) = ax andf2(x) = bx.

Theorem 3.28 Suppose that ξ1, ξ2, · · · , ξn are independent hybrid variables,and f1, f2, · · · , fn are measurable functions. Then f1(ξ1), f2(ξ2), · · · , fn(ξn)are independent hybrid variables.

Proof: The theorem follows from the definition because the compound ofmeasurable functions is also measurable.



This section introduces the concept of identical distribution of hybrid vari-ables.

Definition 3.16 The hybrid variables ξ and η are identically distributed if

Chξ ∈ B = Chη ∈ B (3.44)

for any Borel set B of real numbers.

Theorem 3.29 Let ξ and η be identically distributed hybrid variables, andf : < → < a measurable function. Then f(ξ) and f(η) are identically dis-tributed hybrid variables.

Proof: For any Borel set B of real numbers, we have

Chf(ξ) ∈ B = Chξ ∈ f−1(B) = Chη ∈ f−1(B) = Chf(η) ∈ B.

Hence f(ξ) and f(η) are identically distributed hybrid variables.

Theorem 3.30 If ξ and η are identically distributed hybrid variables, thenthey have the same chance distribution.

Proof: Since ξ and η are identically distributed hybrid variables, we haveChξ ∈ (−∞, x] = Chη ∈ (−∞, x] for any x. Thus ξ and η have the samechance distribution.

Theorem 3.31 If ξ and η are identically distributed hybrid variables whosechance density functions exist, then they have the same chance density func-tion.

Proof: It follows from Theorem 3.30 immediately.

3.9 Critical Values

In order to rank fuzzy random variables, Liu [122] defined two critical values:optimistic value and pessimistic value. Analogously, Liu [124] gave the con-cepts of critical values of random fuzzy variables. Li and Liu [103] presentedthe following definition of critical values of hybrid variables.

Definition 3.17 Let ξ be a hybrid variable, and α ∈ (0, 1]. Then

ξsup(α) = supr∣∣ Ch ξ ≥ r ≥ α

(3.45)


ξinf(α) = infr∣∣ Ch ξ ≤ r ≥ α

(3.46)



The hybrid variable ξ reaches upwards of the α-optimistic value ξsup(α),and is below the α-pessimistic value ξinf(α) with chance α.

Example 3.14: If a hybrid variable ξ degenerates to a random variable η,then

Chξ ≤ x = Prη ≤ x, Chξ ≥ x = Prη ≥ x, ∀x ∈ <.

It follows from the definition of critical values that

ξsup(α) = ηsup(α), ξinf(α) = ηinf(α), ∀α ∈ (0, 1].

In other words, the critical values of hybrid variable coincide with that ofrandom variable.

Example 3.15: If a hybrid variable ξ degenerates to a fuzzy variable a, then

Chξ ≤ x = Cra ≤ x, Chξ ≥ x = Cra ≥ x, ∀x ∈ <.

It follows from the definition of critical values that

ξsup(α) = asup(α), ξinf(α) = ainf(α), ∀α ∈ (0, 1].

In other words, the critical values of hybrid variable coincide with that offuzzy variable.

Theorem 3.32 Let ξ be a hybrid variable. If α > 0.5, then we have

Chξ ≤ ξinf(α) ≥ α, Chξ ≥ ξsup(α) ≥ α. (3.47)

Proof: It follows from the definition of α-pessimistic value that there existsa decreasing sequence xi such that Chξ ≤ xi ≥ α and xi ↓ ξinf(α) asi→∞. Since ξ ≤ xi ↓ ξ ≤ ξinf(α) and limi→∞ Chξ ≤ xi ≥ α > 0.5, itfollows from the chance semicontinuity theorem that

Chξ ≤ ξinf(α) = limi→∞

Chξ ≤ xi ≥ α.

Similarly, there exists an increasing sequence xi such that Chξ ≥xi ≥ α and xi ↑ ξsup(α) as i → ∞. Since ξ ≥ xi ↓ ξ ≥ ξsup(α)and limi→∞ Chξ ≥ xi ≥ α > 0.5, it follows from the chance semicontinuitytheorem that

Chξ ≥ ξsup(α) = limi→∞

Chξ ≥ xi ≥ α.


Theorem 3.33 Let ξ be a hybrid variable and α a number between 0 and 1.We have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).


Proof: (a) If c = 0, then the part (a) is obvious. In the case of c > 0, wehave

(cξ)sup(α) = supr∣∣ Chcξ ≥ r ≥ α

= c supr/c | Chξ ≥ r/c ≥ α

= cξsup(α).

A similar way may prove (cξ)inf(α) = cξinf(α). In order to prove the part (b),it suffices to prove that (−ξ)sup(α) = −ξinf(α) and (−ξ)inf(α) = −ξsup(α).In fact, we have

(−ξ)sup(α) = supr∣∣ Ch−ξ ≥ r ≥ α

= − inf−r | Chξ ≤ −r ≥ α

= −ξinf(α).


Theorem 3.34 Let ξ be a hybrid variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).


1 ≥ Chξ < ξ(α)+ Chξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α).Part (b): Assume that ξinf(α) > ξsup(α). It follows from the definition of

ξinf(α) that Chξ ≤ ξ(α) < α. Similarly, it follows from the definition ofξsup(α) that Chξ ≥ ξ(α) < α. Thus

1 ≤ Chξ ≤ ξ(α)+ Chξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is verified.

Theorem 3.35 Let ξ be a hybrid variable. Then we have(a) ξsup(α) is a decreasing and left-continuous function of α;(b) ξinf(α) is an increasing and left-continuous function of α.

Proof: (a) It is easy to prove that ξinf(α) is an increasing function of α.Next, we prove the left-continuity of ξinf(α) with respect to α. Let αi bean arbitrary sequence of positive numbers such that αi ↑ α. Then ξinf(αi)is an increasing sequence. If the limitation is equal to ξinf(α), then the left-continuity is proved. Otherwise, there exists a number z∗ such that

limi→∞


Thus Chξ ≤ z∗ ≥ αi for each i. Letting i → ∞, we get Chξ ≤ z∗ ≥ α.Hence z∗ ≥ ξinf(α). A contradiction proves the left-continuity of ξinf(α) withrespect to α. The part (b) may be proved similarly.


3.10 Entropy

This section provides a definition of entropy to characterize the uncertaintyof hybrid variables resulting from information deficiency.

Definition 3.18 (Li and Liu [103]) Suppose that ξ is a discrete hybrid vari-able taking values in x1, x2, · · · . Then its entropy is defined by

H[ξ] =∞∑

i=1

S(Chξ = xi) (3.48)


Example 3.16: Suppose that ξ is a discrete hybrid variable taking valuesin x1, x2, · · · . If there exists some index k such that Chξ = xk = 1, and0 otherwise, then its entropy H[ξ] = 0.

Example 3.17: Suppose that ξ is a simple hybrid variable taking values inx1, x2, · · · , xn. If Chξ = xi = 0.5 for all i = 1, 2, · · · , n, then its entropyH[ξ] = n ln 2.

Theorem 3.36 Suppose that ξ is a discrete hybrid variable taking values inx1, x2, · · · . Then

H[ξ] ≥ 0 (3.49)and equality holds if and only if ξ is essentially a deterministic/crisp number.

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifChξ = xi = 0 or 1 for each i. That is, there exists one and only oneindex k such that Chξ = xk = 1, i.e., ξ is essentially a deterministic/crispnumber.

This theorem states that the entropy of a hybrid variable reaches itsminimum 0 when the uncertain variable degenerates to a deterministic/crispnumber. In this case, there is no uncertainty.

Theorem 3.37 Suppose that ξ is a simple hybrid variable taking values inx1, x2, · · · , xn. Then

H[ξ] ≤ n ln 2 (3.50)and equality holds if and only if Chξ = xi = 0.5 for all i = 1, 2, · · · , n.


H[ξ] =n∑

i=1

S(Chξ = xi) ≤ n ln 2

and equality holds if and only if Chξ = xi = 0.5 for all i = 1, 2, · · · , n.

This theorem states that the entropy of a hybrid variable reaches itsmaximum when the hybrid variable is an equipossible one. In this case,there is no preference among all the values that the hybrid variable will take.


3.11 Distance

Definition 3.19 (Li and Liu [103]) The distance between hybrid variables ξand η is defined as

d(ξ, η) = E[|ξ − η|]. (3.51)

Theorem 3.38 (Li and Liu [103]) Let ξ, η, τ be hybrid variables, and letd(·, ·) be the distance. Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ) + 2d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.Now we prove the part (d). It follows from the chance subadditivity theoremthat

d(ξ, η) =∫ +∞

0

Ch |ξ − η| ≥ rdr

≤∫ +∞

0

Ch |ξ − τ |+ |τ − η| ≥ rdr

≤∫ +∞

0

Ch |ξ − τ | ≥ r/2 ∪ |τ − η| ≥ r/2dr

≤∫ +∞

0

(Ch|ξ − τ | ≥ r/2+ Ch|τ − η| ≥ r/2) dr

=∫ +∞

0

Ch|ξ − τ | ≥ r/2dr +∫ +∞

0

Ch|τ − η| ≥ r/2dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ) + 2d(τ, η).

3.12 Inequalities

Yang and Liu [237] proved some important inequalities for fuzzy randomvariables, and Zhu and Liu [262] presented several inequalities for randomfuzzy variables. Li and Liu [103] also verified the following inequalities forhybrid variables.

Theorem 3.39 Let ξ be a hybrid variable, and f a nonnegative function. Iff is even and increasing on [0,∞), then for any given number t > 0, we have

Ch|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (3.52)


Proof: It is clear that Ch|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Chf(ξ) ≥ rdr

=∫ +∞

0

Ch|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

Ch|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr · Ch|ξ| ≥ f−1(f(t))

= f(t) · Ch|ξ| ≥ t


Theorem 3.40 (Markov Inequality) Let ξ be a hybrid variable. Then forany given numbers t > 0 and p > 0, we have

Ch|ξ| ≥ t ≤ E[|ξ|p]tp

. (3.53)


Theorem 3.41 (Chebyshev Inequality) Let ξ be a hybrid variable whose vari-ance V [ξ] exists. Then for any given number t > 0, we have

Ch |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (3.54)

Proof: It is a special case of Theorem 3.39 when the hybrid variable ξ isreplaced with ξ − E[ξ], and f(x) = x2.

Theorem 3.42 (Holder’s Inequality) Let p and q be positive real numberswith 1/p + 1/q = 1, and let ξ and η be independent hybrid variables withE[|ξ|p] <∞ and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (3.55)









E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).


Theorem 3.43 (Minkowski Inequality) Let p be a real number with p ≥1, and let ξ and η be independent hybrid variables with E[|ξ|p] < ∞ andE[|η|p] <∞. Then we have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (3.56)


√x + p







E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Theorem 3.44 (Jensen’s Inequality) Let ξ be a hybrid variable, and f : < →< a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (3.57)



f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).


E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0




Liu [129] gave the convergence concepts of fuzzy random sequence, and Zhuand Liu [264] introduced the convergence concepts of random fuzzy sequence.Li and Liu [103] discussed the convergence concepts of hybrid sequence: con-vergence almost surely (a.s.), convergence in chance, convergence in mean,and convergence in distribution.

Table 3.1: Relationship among Convergence Concepts

Convergence⇒

Convergence⇒

Convergence

in Mean in Chance in Distribution

Definition 3.20 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables defined onthe chance space (Θ,P,Cr) × (Ω,A,Pr). The sequence ξi is said to beconvergent a.s. to ξ if there exists an event Λ with ChΛ = 1 such that

limi→∞

|ξi(θ, ω)− ξ(θ, ω)| = 0 (3.58)

for every (θ, ω) ∈ Λ. In that case we write ξi → ξ, a.s.

Definition 3.21 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables. We say thatthe sequence ξi converges in chance to ξ if

limi→∞

Ch |ξi − ξ| ≥ ε = 0 (3.59)

for every ε > 0.

Definition 3.22 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables with finiteexpected values. We say that the sequence ξi converges in mean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (3.60)


limi→∞

E[|ξi − ξ|2] = 0. (3.61)

Definition 3.23 Suppose that Φ,Φ1,Φ2, · · · are the chance distributions ofhybrid variables ξ, ξ1, ξ2, · · · , respectively. We say that ξi converges in dis-tribution to ξ if Φi → Φ at any continuity point of Φ.


Convergence Almost Surely vs. Convergence in Chance

Example 3.18: Convergence a.s. does not imply convergence in chance.Take a credibility space (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j+1)for j = 1, 2, · · · and take an arbitrary probability space (Ω,A,Pr). Then wedefine hybrid variables as

ξi(θj , ω) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. The sequence ξi convergence a.s. to ξ. However,for some small number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Cr|ξi − ξ| ≥ ε =i

2i+ 1→ 1

2.

That is, the sequence ξi does not converge in chance to ξ.

Example 3.19: Convergence in chance does not imply convergence a.s.Take an arbitrary credibility space (Θ,P,Cr) and take a probability space(Ω,A,Pr) to be [0, 1] with Borel algebra and Lebesgue measure. For anypositive integer i, there is an integer j such that i = 2j + k, where k is aninteger between 0 and 2j − 1. Then we define hybrid variables as

ξi(θ, ω) =

i, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. For some small number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Pr|ξi − ξ| ≥ ε =12i→ 0

as i→∞. That is, the sequence ξi converges in chance to ξ. However, forany ω ∈ [0, 1], there is an infinite number of intervals of the form [k/2j , (k +1)/2j ] containing ω. Thus ξi(θ, ω) does converges to 0. In other words, thesequence ξi does not converge a.s. to ξ.

Convergence in Mean vs. Convergence in Chance

Theorem 3.45 Suppose that ξ, ξ1, ξ2, · · · are hybrid variables. If ξi con-verges in mean to ξ, then ξi converges in chance to ξ.

Proof: It follows from the Markov inequality that for any given numberε > 0, we have

Ch|ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in chance to ξ. The theorem is proved.


Example 3.20: Convergence in chance does not imply convergence in mean.Take a credibility space (Θ,P,Cr) to be θ1, θ2, · · · with Crθ1 = 1/2and Crθj = 1/j for j = 2, 3, · · · and take an arbitrary probability space(Ω,A,Pr). The hybrid variables are defined by

ξi(θj , ω) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. For some small number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Cr|ξi − ξ| ≥ ε =1i→ 0.

That is, the sequence ξi converges in chance to ξ. However,

E[|ξi − ξ|] = 1, ∀i.


Convergence Almost Surely vs. Convergence in Mean

Example 3.21: Convergence a.s. does not imply convergence in mean.Take an arbitrary credibility space (Θ,P,Cr) and take a probability space(Ω,A,Pr) to be ω1, ω2, · · · with Prωj = 1/2j for j = 1, 2, · · · The hybridvariables are defined by

ξi(θ, ωj) =

2i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. Then ξi converges a.s. to ξ. However, the sequenceξi does not converges in mean to ξ because E[|ξi − ξ|] ≡ 1.

Example 3.22: Convergence in chance does not imply convergence a.s.Take an arbitrary credibility space (Θ,P,Cr) and take a probability space(Ω,A,Pr) to be [0, 1] with Borel algebra and Lebesgue measure. For anypositive integer i, there is an integer j such that i = 2j + k, where k is aninteger between 0 and 2j − 1. The hybrid variables are defined by

ξi(θ, ω) =

i, if k/2j ≤ ω ≤ (k + 1)/2j

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. Then

E[|ξi − ξ|] =12j→ 0.

That is, the sequence ξi converges in mean to ξ. However, for any ω ∈ [0, 1],there is an infinite number of intervals of the form [k/2j , (k+1)/2j ] containingω. Thus ξi(θ, ω) does converges to 0. In other words, the sequence ξi doesnot converge a.s. to ξ.


Convergence in Chance vs. Convergence in Distribution

Theorem 3.46 Suppose ξ, ξ1, ξ2, · · · are hybrid variables. If ξi convergesin chance to ξ, then ξi converges in distribution to ξ.

Proof: Let x be a given continuity point of the distribution Φ. On the onehand, for any y > x, we have


It follows from the chance subadditivity theorem that

Φi(x) ≤ Φ(y) + Ch|ξi − ξ| ≥ y − x.

Since ξi converges in chance to ξ, we have Ch|ξi − ξ| ≥ y − x → 0 asi → ∞. Thus we obtain lim supi→∞ Φi(x) ≤ Φ(y) for any y > x. Lettingy → x, we get

lim supi→∞

Φi(x) ≤ Φ(x). (3.62)


ξ ≤ z = ξi ≤ x, ξ ≤ z ∪ ξi > x, ξ ≤ z ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) + Ch|ξi − ξ| ≥ x− z.

Since Ch|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞ Φi(x) for anyz < x. Letting z → x, we get


Φi(x). (3.63)


Example 3.23: Convergence in distribution does not imply convergencein chance. Take a credibility space (Θ,P,Cr) to be θ1, θ2 with Crθ1 =Crθ2 = 1/2 and take an arbitrary probability space (Ω,A,Pr). We definea hybrid variables as

ξ(θ, ω) =

−1, if θ = θ1

1, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · . Then ξi and ξ have the same chancedistribution. Thus ξi converges in distribution to ξ. However, for somesmall number ε > 0, we have

Ch|ξi − ξ| ≥ ε = Cr|ξi − ξ| ≥ ε = 1.

That is, the sequence ξi does not converge in chance to ξ.


Convergence Almost Surely vs. Convergence in Distribution

Example 3.24: Convergence in distribution does not imply convergencea.s. Take a credibility space to be (Θ,P,Cr) to be θ1, θ2 with Crθ1 =Crθ2 = 1/2 and take an arbitrary probability space (Ω,A,Pr). We definea hybrid variable ξ as

ξ(θ, ω) =

−1, if θ = θ1

1, if θ = θ2.

We also define ξi = −ξ for i = 1, 2, · · · . Then ξi and ξ have the same chancedistribution. Thus ξi converges in distribution to ξ. However, the sequenceξi does not converge a.s. to ξ.

Example 3.25: Convergence a.s. does not imply convergence in chance.Take a credibility space (Θ,P,Cr) to be θ1, θ2, · · · with Crθj = j/(2j+1)for j = 1, 2, · · · and take an arbitrary probability space (Ω,A,Pr). The hybridvariables are defined by

ξi(θj , ω) =

i, if j = i

0, otherwise

for i = 1, 2, · · · and ξ ≡ 0. Then the sequence ξi converges a.s. to ξ.However, the chance distributions of ξi are

Φi(x) =

0, if x < 0

(i+ 1)/(2i+ 1), if 0 ≤ x < i

1, if x ≥ i

for i = 1, 2, · · · , respectively. The chance distribution of ξ is

Φ(x) =

0, if x < 01, if x ≥ 0.

It is clear that Φi(x) does not converge to Φ(x) at x > 0. That is, thesequence ξi does not converge in distribution to ξ.

3.14 Conditional Chance

We consider the chance measure of an event A after it has been learned thatsome other event B has occurred. This new chance measure of A is calledthe conditional chance measure of A given B.

In order to define a conditional chance measure ChA|B, at first wehave to enlarge ChA ∩ B because ChA ∩ B < 1 for all events wheneverChB < 1. It seems that we have no alternative but to divide ChA∩B by

Section 3.14 - Conditional Chance 165

ChB. Unfortunately, ChA ∩B/ChB is not always a chance measure.However, the value ChA|B should not be greater than ChA∩B/ChB(otherwise the normality will be lost), i.e.,

ChA|B ≤ ChA ∩BChB

. (3.64)


ChA|B = 1− ChAc|B ≥ 1− ChAc ∩BChB

. (3.65)

Furthermore, since (A∩B)∪ (Ac ∩B) = B, we have ChB ≤ ChA∩B+ChAc ∩B by using the chance subadditivity theorem. Thus

0 ≤ 1− ChAc ∩BChB

≤ ChA ∩BChB

≤ 1. (3.66)

Hence any numbers between 1−ChAc∩B/ChB and ChA∩B/ChBare reasonable values that the conditional chance may take. Based on themaximum uncertainty principle, we have the following conditional chancemeasure.

Definition 3.24 (Li and Liu [106]) Let (Θ,P,Cr) × (Ω,A,Pr) be a chancespace and A,B two events. Then the conditional chance measure of A givenB is defined by

ChA|B =

ChA ∩BChB

, ifChA ∩B

ChB< 0.5

1− ChAc ∩BChB

, ifChAc ∩B

ChB< 0.5

0.5, otherwise

(3.67)

provided that ChB > 0.

Remark 3.6: It follows immediately from the definition of conditionalchance that

1− ChAc ∩BChB

≤ ChA|B ≤ ChA ∩BChB

. (3.68)

Furthermore, it is clear that the conditional chance measure obeys the max-imum uncertainty principle.


Remark 3.7: Let X and Y be events in the credibility space. Then theconditional chance measure of X × Ω given Y × Ω is

ChX × Ω|Y × Ω =

CrX ∩ Y CrY

, ifCrX ∩ Y

CrY < 0.5

1− CrXc ∩ Y CrY

, ifCrXc ∩ Y

CrY < 0.5

0.5, otherwise

which is just the conditional credibility of X given Y .

Remark 3.8: Let X and Y be events in the probability space. Then theconditional chance measure of Θ×X given Θ× Y is

ChΘ×X|Θ× Y =PrX ∩ Y

PrY

which is just the conditional probability of X given Y .

Example 3.26: Let ξ and η be two hybrid variables. Then we have

Ch ξ = x|η = y=

Chξ = x, η = yChη = y

, ifChξ = x, η = y

Chη = y< 0.5

1− Chξ 6= x, η = yChη = y

, ifChξ 6= x, η = y

Chη = y< 0.5

0.5, otherwise

provided that Chη = y > 0.

Theorem 3.47 (Li and Liu [106]) Conditional chance measure is a type ofuncertain measure. That is, conditional chance measure is normal, increas-ing, self-dual and countably subadditive.

Proof: At first, the conditional chance measure Ch·|B is normal, i.e.,

ChΘ× Ω|B = 1− Ch∅ChB

= 1.


ChA1 ∩BChB

≤ ChA2 ∩BChB

< 0.5,

then

ChA1|B =ChA1 ∩B

ChB≤ ChA2 ∩B

ChB= ChA2|B.

Section 3.14 - Conditional Chance 167

IfChA1 ∩B

ChB≤ 0.5 ≤ ChA2 ∩B

ChB,

then ChA1|B ≤ 0.5 ≤ ChA2|B. If

0.5 <ChA1 ∩B

ChB≤ ChA2 ∩B

ChB,

then we have

ChA1|B =(1− ChAc

1 ∩BChB

)∨0.5 ≤

(1− ChAc

2 ∩BChB

)∨0.5 = ChA2|B.

This means that Ch·|B is increasing. For any event A, if

ChA ∩BChB

≥ 0.5,ChAc ∩B

ChB≥ 0.5,

then we have ChA|B+ChAc|B = 0.5+0.5 = 1 immediately. Otherwise,without loss of generality, suppose

ChA ∩BChB

< 0.5 <ChAc ∩B

ChB,

then we have

ChA|B+ ChAc|B =ChA ∩B

ChB+(

1− ChA ∩BChB

)= 1.

That is, Ch·|B is self-dual. Finally, for any countable sequence Ai ofevents, if ChAi|B < 0.5 for all i, it follows from the countable subadditivityof chance measure that

Ch

∞⋃i=1

Ai ∩B

≤

Ch

∞⋃i=1

Ai ∩B

ChB

≤

∞∑i=1

ChAi ∩B

ChB=

∞∑i=1

ChAi|B.

Suppose there is one term greater than 0.5, say

ChA1|B ≥ 0.5, ChAi|B < 0.5, i = 2, 3, · · ·

If Ch∪iAi|B = 0.5, then we immediately have

Ch

∞⋃i=1

Ai ∩B

≤

∞∑i=1

ChAi|B.

If Ch∪iAi|B > 0.5, we may prove the above inequality by the followingfacts:

Ac1 ∩B ⊂

∞⋃i=2

(Ai ∩B) ∪

( ∞⋂i=1

Aci ∩B

),


ChAc1 ∩B ≤

∞∑i=2

ChAi ∩B+ Ch

∞⋂i=1

Aci ∩B

,

Ch

∞⋃i=1

Ai|B

= 1−

Ch

∞⋂i=1

Aci ∩B

ChB

,

∞∑i=1

ChAi|B ≥ 1− ChAc1 ∩B

ChB+

∞∑i=2

ChAi ∩B

ChB.

If there are at least two terms greater than 0.5, then the countable subaddi-tivity is clearly true. Thus Ch·|B is countably subadditive. Hence Ch·|Bis an uncertain measure.

Definition 3.25 (Li and Liu [106]) The conditional chance distribution Φ:< → [0, 1] of a hybrid variable ξ given B is defined by

Φ(x|B) = Ch ξ ≤ x|B (3.69)

provided that ChB > 0.

Example 3.27: Let ξ and η be hybrid variables. Then the conditionalchance distribution of ξ given η = y is

Φ(x|η = y) =

Chξ ≤ x, η = yChη = y

, ifChξ ≤ x, η = y

Chη = y< 0.5

1− Chξ > x, η = yChη = y

, ifChξ > x, η = y

Chη = y< 0.5

0.5, otherwise

provided that Chη = y > 0.

Definition 3.26 (Li and Liu [106]) The conditional chance density functionφ of a hybrid variable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ <, (3.70)

∫ +∞

−∞φ(y|B)dy = 1 (3.71)

where Φ(x|B) is the conditional chance distribution of ξ given B.

Section 3.15 - Hybrid Process 169

Definition 3.27 (Li and Liu [106]) Let ξ be a hybrid variable. Then theconditional expected value of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Chξ ≥ r|Bdr −∫ 0

−∞Chξ ≤ r|Bdr (3.72)


Following conditional chance and conditional expected value, we also haveconditional variance, conditional moments, conditional critical values, condi-tional entropy as well as conditional convergence.

3.15 Hybrid Process

Definition 3.28 (Liu [133]) Let T be an index set, and (Θ,P,Cr)×(Ω,A,Pr)a chance space. A hybrid process is a measurable function from T×(Θ,P,Cr)×(Ω,A,Pr) to the set of real numbers, i.e., for each t ∈ T and any Borel setB of real numbers, the set

(θ, ω) ∈ Θ× Ω∣∣ X(t, θ, ω) ∈ B (3.73)

is an event.

That is, a hybrid process X(θ, ω) is a function of three variables suchthat the function Xt∗(θ, ω) is a hybrid variable for each t∗. For each fixed(θ∗, ω∗), the function Xt(θ∗, ω∗) is called a sample path of the hybrid process.A hybrid process Xt(θ, ω) is said to be sample-continuous if the sample pathis continuous for almost all (θ, ω).

Definition 3.29 (Liu [133]) A hybrid process Xt is said to have independentincrements if

Xt1 −Xt0 , Xt2 −Xt1 , · · · , Xtk−Xtk−1 (3.74)

are independent hybrid variables for any times t0 < t1 < · · · < tk. A hybridprocess Xt is said to have stationary increments if, for any given t > 0, theXs+t −Xs are identically distributed hybrid variables for all s > 0.

Example 3.28: Let Xt be a fuzzy process and let Yt be a stochastic process.Then Xt + Yt is a hybrid process.

Hybrid Renewal Process

Definition 3.30 (Liu [133]) Let ξ1, ξ2, · · · be iid positive hybrid variables.Define S0 = 0 and Sn = ξ1 + ξ2 + · · ·+ ξn for n ≥ 1. Then the hybrid process

Nt = maxn≥0

n∣∣ Sn ≤ t

(3.75)

is called a hybrid renewal process.




ChNt ≥ n = ChSn ≤ t. (3.76)

Theorem 3.48 (Liu [133]) Let Nt be a hybrid renewal process. Then wehave

E[Nt] =∞∑

n=1

ChSn ≤ t. (3.77)


E[Nt] =∫ ∞

0

ChNt ≥ rdr =∞∑

n=1

∫ n

n−1

ChNt ≥ rdr

=∞∑

n=1

ChNt ≥ n =∞∑

n=1

ChSn ≤ t.


D Process

Definition 3.31 (Liu [133]) Let Bt be a Brownian motion, and let Ct be aC process. Then

Dt = (Bt, Ct) (3.78)

is called a D process. The D process is said to be standard if both Bt and Ct

are standard.

Definition 3.32 (Liu [133]) Let Bt be a standard Brownian motion, and letCt be a standard C process. Then the hybrid process

D∗t = et+ σ1Bt + σ2Ct (3.79)

is called a scalar D process. The parameter e is called the drift coefficient, σ1

is called the random diffusion coefficient, and σ2 is called the fuzzy diffusioncoefficient.

Definition 3.33 (Liu [133]) Let Bt be a standard Brownian motion, and letCt be a standard C process. Then the hybrid process

Gt = exp(et+ σ1Bt + σ2Ct) (3.80)

is called a geometric D process.

Section 3.16 - Hybrid Calculus 171

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Figure 3.2: A Sample Path of Standard D Process

A Hybrid Stock Model

Liu [133] assumed that stock price follows geometric D process, and presenteda hybrid stock model in which the bond price Xt and the stock price Yt aredetermined by

Xt = X0 exp(rt)

Yt = Y0 exp(et+ σ1Bt + σ2Ct)(3.81)

where r is the riskless interest rate, e is the stock drift, σ1 is the randomstock diffusion, σ2 is the fuzzy stock diffusion, Bt is a standard Brownianmotion, and Ct is a standard C process. For exploring further developmentof hybrid stock models, the interested readers may consult Gao [51].

3.16 Hybrid Calculus

Let Dt be a standard D process, and dt an infinitesimal time interval. Then

dDt = Dt+dt −Dt = (dBt,dCt)

is a hybrid process.

Definition 3.34 (Liu [133]) Let Xt = (Yt, Zt) where Yt and Zt are scalarhybrid processes, and let Dt = (Bt, Ct) be a standard D process. For anypartition of closed interval [a, b] with a = t1 < t2 < · · · < tk+1 = b, the meshis written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the hybrid integral of Xt with respect to Dt is∫ b

a

XtdDt = lim∆→0

k∑i=1

(Yti(Bti+1 −Bti) + Zti(Cti+1 − Cti)

)(3.82)


provided that the limit exists in mean square and is a hybrid variable.

Remark 3.9: The hybrid integral may also be written as follows,∫ b

a

XtdDt =∫ b

a

(YtdBt + ZtdCt) . (3.83)

Example 3.29: Let Bt be a standard Brownian motion, and Ct a standardC process. Then ∫ s

0

(σ1dBt + σ2dCt) = σ1Bs + σ2Cs

where σ1 and σ2 are constants, random variables, fuzzy variables, or hybridvariables.

Example 3.30: Let Bt be a standard Brownian motion, and Ct a standardC process. Then ∫ s

0

(BtdBt + CtdCt) =12(B2

s − s+ C2s ).

Example 3.31: Let Bt be a standard Brownian motion, and Ct a standardC process. Then for any partition 0 = t1 < t2 < · · · < tk+1 = s, write

∆Bi = Bti+1 −Bti, ∆Ci = Cti+1 − Cti

and obtain

BsCs =k∑

i=1

(Bti+1Cti+1 −Bti

Cti

)=

k∑i=1

Bti∆Ci +k∑

i=1

Cti∆Bi +k∑

i=1

∆Bi∆Ci

→∫ s

0

BtdCt +∫ s

0

CtdBt + 0


0

(CtdBt +BtdCt) = BsCs.

Theorem 3.49 (Liu [133]) Let Bt be a standard Brownian motion, Ct astandard C process, and h(t, b, c) a twice continuously differentiable function.Define Xt = h(t, Bt, Ct). Then we have the following chain rule

dXt =∂h

∂t(t, Bt, Ct)dt+

∂h

∂b(t, Bt, Ct)dBt

+∂h

∂c(t, Bt, Ct)dCt +

12∂2h

∂b2(t, Bt, Ct)dt.

(3.84)

Section 3.17 - Hybrid Differential Equation 173


∆Xt =∂h

∂t(t, Bt, Ct)∆t+

∂h

∂b(t, Bt, Ct)∆Bt +

∂h

∂c(t, Bt, Ct)∆Ct

+12∂2h

∂t2(t, Bt, Ct)(∆t)2 +

12∂2h

∂b2(t, Bt, Ct)(∆Bt)2 +

12∂2h

∂c2(t, Bt, Ct)(∆Ct)2

+∂2h

∂t∂b(t, Bt, Ct)∆t∆Bt +

∂2h

∂t∂c(t, Bt, Ct)∆t∆Ct +

∂2h

∂b∂c(t, Bt, Ct)∆Bt∆Ct.

Since we can ignore the terms (∆t)2, (∆Ct)2, ∆t∆Bt, ∆t∆Ct, ∆Bt∆Ct andreplace (∆Bt)2 with ∆t, the chain rule is obtained because it makes

Xs = X0 +∫ s

0

∂h

∂tdt+

∫ s

0

∂h

∂bdBt +

∫ s

0

∂h

∂cdCt +

12

∫ s

0

∂2h

∂b2dt

for any s ≥ 0.

Remark 3.10: The infinitesimal increments dBt and dCt in (3.84) may bereplaced with the derived D process

dYt = utdt+ v1tdBt + v2tdCt (3.85)

where ut and v2t are absolutely integrable hybrid processes, and v1t is asquare integrable hybrid process, thus producing

dh(t, Yt) =∂h

∂t(t, Yt)dt+

∂h

∂b(t, Yt)dYt +

12∂2h

∂b2(t, Yt)v2

1tdt. (3.86)

Remark 3.11: Assume that B1t, B2t, · · · , Bmt are standard Brownian mo-tions, C1t, C2t, · · · , Cnt are standard C processes, and

h(t, b1, b2, · · · , bm, c1, c2, · · · , cn)

is a twice continuously differentiable function. Define

Xt = h(t, B1t, B2t, · · · , Bmt, C1t, C2t, · · · , Cnt).

You [244] proved the following chain rule

dXt =∂h

∂tdt+

m∑i=1

∂h

∂bidBit +

n∑j=1

∂h

∂cjdCjt +

12

m∑i=1

∂2h

∂b2idt. (3.87)

Example 3.32: Applying the chain rule, we obtain the following formulas,

d(BtCt) = CtdBt +BtdCt,

d(tBtCt) = BtCtdt+ tCtdBt + tBtdCt,

d(t2 +B2t + C2

t ) = 2tdt+ 2BtdBt + 2CtdCt + dt.


3.17 Hybrid Differential Equation

Hybrid differential equation is a type of differential equation driven by bothBrownian motion and C process.

Definition 3.35 (Liu [133]) Suppose Bt is a standard Brownian motion, Ct

is a standard C process, and f, g1, g2 are some given functions. Then

dXt = f(t,Xt)dt+ g1(t,Xt)dBt + g2(t,Xt)dCt (3.88)

is called a hybrid differential equation. A solution is a hybrid process Xt thatsatisfies (3.88) identically in t.

Remark 3.12: Note that there is no precise definition for the terms dXt, dt,dBt and dCt in the hybrid differential equation (3.88). The mathematicallymeaningful form is the hybrid integral equation

Xs = X0 +∫ s

0

f(t,Xt)dt+∫ s

0

g1(t,Xt)dBt +∫ s

0

g2(t,Xt)dCt. (3.89)


Example 3.33: Let Bt be a standard Brownian motion, and let a and b betwo fuzzy variables. Then the hybrid differential equation

dXt = adt+ bdBt

has a solutionXt = at+ bBt.

The hybrid differential equation

dXt = aXtdt+ bXtdBt

has a solution

Xt = exp

((a− b2

2

)t+ bBt

).

Example 3.34: Let Ct be a standard C process, and let ξ and η be tworandom variables. Then the hybrid differential equation

dXt = ξdt+ ηdCt

has a solutionXt = ξt+ ηCt.

Section 3.17 - Hybrid Differential Equation 175

The hybrid differential equation

dXt = ξXtdt+ ηXtdCt

has a solutionXt = exp (ξt+ ηCt) .

Example 3.35: Let Bt be a standard Brownian motion, and Ct a standardC process. Then the hybrid differential equation

dXt = adt+ bdBt + cdCt

has a solutionXt = at+ bBt + cCt

which is just a scalar D process.

Example 3.36: Let Bt be a standard Brownian motion, and Ct a standardC process. Then the hybrid differential equation

dXt = aXtdt+ bXtdBt + cXtdCt

has a solution

Xt = exp((

a− b2

2

)t+ bBt + cCt

)which is just a geometric D process.

Chapter 4

Uncertainty Theory

A classical measure is essentially a set function satisfying nonnegativity andcountable additivity axioms. Classical measure theory, developed by EmileBorel and Henri Lebesgue around 1900, has been widely applied in boththeory and practice.

However, the additivity axiom of classical measure theory has been chal-lenged by many mathematicians. The earliest challenge was from the theoryof capacities by Choquet [22] in which monotonicity and continuity axiomswere assumed. Sugeno [214] generalized classical measure theory to fuzzymeasure theory by replacing additivity axiom with monotonicity and semi-continuity axioms.

In order to deal with general uncertainty, “self-duality” plus “countablesubadditivity” is much more important than “continuity” and “semicontinu-ity”. For this reason, Liu [132] founded an uncertainty theory that is a branchof mathematics based on normality, monotonicity, self-duality, and countablesubadditivity axioms. Uncertainty theory provides the commonness of prob-ability theory, credibility theory and chance theory.

The emphasis in this chapter is mainly on uncertain measure, uncertaintyspace, uncertain variable, uncertainty distribution, expected value, variance,moments, independence, identical distribution, critical values, entropy, dis-tance, convergence almost surely, convergence in measure, convergence inmean, convergence in distribution, conditional uncertainty, uncertain process,canonical process, uncertain calculus, and uncertain differential equation.

4.1 Uncertainty Space

Let Γ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ ∈ Lis called an event. In order to present an axiomatic definition of uncertainmeasure, it is necessary to assign to each event Λ a number MΛ whichindicates the level that Λ will occur. In order to ensure that the number

178 Chapter 4 - Uncertainty Theory

MΛ has certain mathematical properties, Liu [132] proposed the followingfour axioms:

Axiom 1. (Normality) MΓ = 1.

Axiom 2. (Monotonicity) MΛ1 ≤MΛ2 whenever Λ1 ⊂ Λ2.

Axiom 3. (Self-Duality) MΛ+MΛc = 1 for any event Λ.

Axiom 4. (Countable Subadditivity) For every countable sequence of eventsΛi, we have

M

∞⋃i=1

Λi

≤

∞∑i=1

MΛi. (4.1)

.

Remark 4.1: Pathology occurs if self-duality axiom is not assumed. Forexample, we define a set function that takes value 1 for each set. Then itsatisfies all axioms but self-duality. Is it not strange if such a set functionserves as a measure?

Remark 4.2: Pathology occurs if subadditivity is not assumed. For ex-ample, suppose that a universal set contains 3 elements. We define a setfunction that takes value 0 for each singleton, and 1 for each set with at least2 elements. Then such a set function satisfies all axioms but subadditivity.Is it not strange if such a set function serves as a measure?

Remark 4.3: Pathology occurs if countable subadditivity axiom is replacedwith finite subadditivity axiom. For example, assume the universal set con-sists of all real numbers. We define a set function that takes value 0 if theset is bounded, 0.5 if both the set and complement are unbounded, and 1 ifthe complement of the set is bounded. Then such a set function is finitelysubadditive but not countably subadditive. Is it not strange if such a setfunction serves as a measure?

Definition 4.1 (Liu [132]) The set function M is called an uncertain mea-sure if it satisfies the normality, monotonicity, self-duality, and countablesubadditivity axioms.

Example 4.1: Probability, credibility and chance measures are instances ofuncertain measure.

Example 4.2: Let Pr be a probability measure, Cr a credibility measure,and a a number in [0, 1]. Then

MΛ = aPrΛ+ (1− a)CrΛ (4.2)

is an uncertain measure.

Section 4.1 - Uncertainty Space 179

Example 4.3: Let Γ = γ1, γ2, γ3. For this case, there are only 8 events.Define

Mγ1 = 0.6, Mγ2 = 0.3, Mγ3 = 0.2,

Mγ1, γ2 = 0.8, Mγ1, γ3 = 0.7, Mγ2, γ3 = 0.4,

M∅ = 0, MΓ = 1.

Then M is an uncertain measure because it satisfies the four axioms.

Example 4.4: Let Γ = [−1, 1] with Borel algebra L and Lebesgue measureπ. Then the set function

MΛ =

πΛ, if πΛ < 0.5

1− πΛc, if πΛc < 0.5

0.5, otherwise

is an uncertain measure.

Theorem 4.1 Suppose that M is an uncertain measure. Then we have

M∅ = 0, (4.3)

0 ≤MΛ ≤ 1 (4.4)

for any event Λ.

Proof: It follows from the normality and self-duality axioms that M∅ =1 −MΓ = 1 − 1 = 0. It follows from the monotonicity axiom that 0 ≤MΛ ≤ 1 because ∅ ⊂ Λ ⊂ Γ.

Theorem 4.2 Suppose thatM is an uncertain measure. Then for any eventsΛ1 and Λ2, we have

MΛ1 ∨MΛ2 ≤MΛ1 ∪ Λ2 ≤MΛ1+MΛ2 (4.5)

Proof: The left-hand inequality follows from the monotonicity axiom andthe right-hand inequality follows from the countable subadditivity axiom im-mediately.

Theorem 4.3 Let Γ = γ1, γ2, · · · . If M is an uncertain measure, then

Mγi+Mγj ≤ 1 ≤∞∑

k=1

Mγk (4.6)

for any i and j.


Proof: Since M is increasing and self-dual, we have, for any i and j,

Mγi+Mγj ≤MΓ\γj+Mγj = 1.

Since Γ = ∪kγk and M is countably subadditive, we have

1 = MΓ = M

∞⋃k=1

γk

≤

∞∑k=1

Mγk.


Uncertainty Null-Additivity Theorem

Null-additivity is a direct deduction from subadditivity. We first prove amore general theorem.

Theorem 4.4 Let Λi be a decreasing sequence of events with MΛi → 0as i→∞. Then for any event Λ, we have

limi→∞

MΛ ∪ Λi = limi→∞

MΛ\Λi = MΛ. (4.7)

Proof: It follows from the monotonicity and countable subadditivity axiomsthat

MΛ ≤MΛ ∪ Λi ≤MΛ+MΛi

for each i. Thus we get MΛ ∪ Λi → MΛ by using MΛi → 0. Since(Λ\Λi) ⊂ Λ ⊂ ((Λ\Λi) ∪ Λi), we have

MΛ\Λi ≤MΛ ≤MΛ\Λi+MΛi.

Hence MΛ\Λi →MΛ by using MΛi → 0.

Remark 4.4: It follows from the above theorem that the uncertain measureis null-additive, i.e., MΛ1 ∪ Λ2 = MΛ1 + MΛ2 if either MΛ1 = 0or MΛ2 = 0. In other words, the uncertain measure remains unchanged ifthe event is enlarged or reduced by an event with measure zero.

Uncertainty Asymptotic Theorem

Theorem 4.5 (Uncertainty Asymptotic Theorem) For any events Λ1,Λ2, · · · ,we have

limi→∞

MΛi > 0, if Λi ↑ Γ, (4.8)

limi→∞

MΛi < 1, if Λi ↓ ∅. (4.9)

Section 4.2 - Uncertain Variables 181

Proof: Assume Λi ↑ Γ. Since Γ = ∪iΛi, it follows from the countablesubadditivity axioms that

1 = MΓ ≤∞∑

i=1

MΛi.

Since MΛi is increasing with respect to i, we have limi→∞MΛi > 0. IfΛi ↓ ∅, then Λc

i ↑ Γ. It follows from the first inequality and self-duality axiomthat

limi→∞

MΛi = 1− limi→∞

MΛci < 1.


Example 4.5: Assume Γ is the set of real numbers. Let α be a number with0 < α ≤ 0.5. Define a set function as follows,

MΛ =

0, if Λ = ∅α, if Λ is upper bounded0.5, if both Λ and Λc are upper unbounded

1− α, if Λc is upper bounded1, if Λ = Γ.

(4.10)

It is easy to verify that M is an uncertain measure. Write Λi = (−∞, i] fori = 1, 2, · · · Then Λi ↑ Γ and limi→∞MΛi = α. Furthermore, we haveΛc

i ↓ ∅ and limi→∞MΛci = 1− α.

Uncertainty Space

Definition 4.2 (Liu [132]) Let Γ be a nonempty set, L a σ-algebra overΓ, and M an uncertain measure. Then the triplet (Γ,L,M) is called anuncertainty space.

4.2 Uncertain Variables

Definition 4.3 (Liu [132]) An uncertain variable is a measurable functionfrom an uncertainty space (Γ,L,M) to the set of real numbers, i.e., for anyBorel set B of real numbers, the set

ξ ∈ B = γ ∈ Γ∣∣ ξ(γ) ∈ B (4.11)

is an event.

Example 4.6: Random variable, fuzzy variable and hybrid variable areinstances of uncertain variable.


Definition 4.4 An uncertain variable ξ on the uncertainty space (Γ,L,M)is said to be(a) nonnegative if Mξ < 0 = 0;(b) positive if Mξ ≤ 0 = 0;(c) continuous if Mξ = x is a continuous function of x;(d) simple if there exists a finite sequence x1, x2, · · · , xm such that

M ξ 6= x1, ξ 6= x2, · · · , ξ 6= xm = 0; (4.12)


M ξ 6= x1, ξ 6= x2, · · · = 0. (4.13)

It is clear that 0 ≤Mξ = x ≤ 1, and there is at most one point x0 suchthat Mξ = x0 > 0.5. For a continuous uncertain variable, we always have0 ≤Mξ = x ≤ 0.5.

Definition 4.5 Let ξ1 and ξ2 be uncertain variables defined on the uncer-tainty space (Γ,L,M). We say ξ1 = ξ2 if ξ1(γ) = ξ2(γ) for almost all γ ∈ Γ.

Uncertain Vector

Definition 4.6 An n-dimensional uncertain vector is a measurable functionfrom an uncertainty space (Γ,L,M) to the set of n-dimensional real vectors,i.e., for any Borel set B of <n, the set

ξ ∈ B =γ ∈ Γ

∣∣ ξ(γ) ∈ B

(4.14)

is an event.

Theorem 4.6 The vector (ξ1, ξ2, · · · , ξn) is an uncertain vector if and onlyif ξ1, ξ2, · · · , ξn are uncertain variables.

Proof: Write ξ = (ξ1, ξ2, · · · , ξn). Suppose that ξ is an uncertain vector onthe uncertainty space (Γ,L,M). For any Borel set B of <, the set B ×<n−1

is a Borel set of <n. Thus the setγ ∈ Γ

∣∣ ξ1(γ) ∈ B=γ ∈ Γ

∣∣ ξ1(γ) ∈ B, ξ2(γ) ∈ <, · · · , ξn(γ) ∈ <

=γ ∈ Γ

∣∣ ξ(γ) ∈ B ×<n−1

is an event. Hence ξ1 is an uncertain variable. A similar process mayprove that ξ2, ξ3, · · · , ξn are uncertain variables. Conversely, suppose thatall ξ1, ξ2, · · · , ξn are uncertain variables on the uncertainty space (Γ,L,M).We define

B =B ⊂ <n

∣∣ γ ∈ Γ|ξ(γ) ∈ B is an event.

Section 4.2 - Uncertain Variables 183

The vector ξ = (ξ1, ξ2, · · · , ξn) is proved to be an uncertain vector if we canprove that B contains all Borel sets of <n. First, the class B contains allopen intervals of <n because

γ∣∣ ξ(γ) ∈

n∏i=1

(ai, bi)

=

n⋂i=1

γ∣∣ ξi(γ) ∈ (ai, bi)

is an event. Next, the class B is a σ-algebra of <n because (i) we have <n ∈ Bsince γ|ξ(γ) ∈ <n = Γ; (ii) if B ∈ B, then

γ ∈ Γ∣∣ ξ(γ) ∈ B

is an event, and

γ ∈ Γ∣∣ ξ(γ) ∈ Bc = γ ∈ Γ

∣∣ ξ(γ) ∈ Bc

is an event. This means that Bc ∈ B; (iii) if Bi ∈ B for i = 1, 2, · · · , thenγ ∈ Γ|ξ(γ) ∈ Bi are events and

γ ∈ Γ∣∣ ξ(γ) ∈

∞⋃i=1

Bi

=

∞⋃i=1

γ ∈ Γ∣∣ ξ(γ) ∈ Bi

is an event. This means that ∪iBi ∈ B. Since the smallest σ-algebra con-taining all open intervals of <n is just the Borel algebra of <n, the class Bcontains all Borel sets of <n. The theorem is proved.

Uncertain Arithmetic

Definition 4.7 Suppose that f : <n → < is a measurable function, andξ1, ξ2, · · · , ξn uncertain variables on the uncertainty space (Γ,L,M). Thenξ = f(ξ1, ξ2, · · · , ξn) is an uncertain variable defined as

ξ(γ) = f(ξ1(γ), ξ2(γ), · · · , ξn(γ)), ∀γ ∈ Γ. (4.15)

Example 4.7: Let ξ1 and ξ2 be two uncertain variables. Then the sumξ = ξ1 + ξ2 is an uncertain variable defined by

ξ(γ) = ξ1(γ) + ξ2(γ), ∀γ ∈ Γ.

The product ξ = ξ1ξ2 is also an uncertain variable defined by

ξ(γ) = ξ1(γ) · ξ2(γ), ∀γ ∈ Γ.

The reader may wonder whether ξ(γ1, γ2, · · · , γn) defined by (4.15) is anuncertain variable. The following theorem answers this question.

Theorem 4.7 Let ξ be an n-dimensional uncertain vector, and f : <n → <a measurable function. Then f(ξ) is an uncertain variable.


Proof: Assume that ξ is an uncertain vector on the uncertainty space(Γ,L,M). For any Borel set B of <, since f is a measurable function, thef−1(B) is a Borel set of <n. Thus the set

γ ∈ Γ∣∣ f(ξ(γ)) ∈ B

=γ ∈ Γ

∣∣ ξ(γ) ∈ f−1(B)

is an event for any Borel set B. Hence f(ξ) is an uncertain variable.

4.3 Identification Function

As we know, a random variable may be characterized by a probability densityfunction, and a fuzzy variable may be described by a membership function.This section will introduce an identification function to characterize an un-certain variable.

Definition 4.8 An uncertain variable ξ is said to have an identificationfunction (λ, ρ) if(i) λ(x) is a nonnegative function and ρ(x) is a nonnegative and integrablefunction such that

supx∈<

λ(x) +∫<ρ(x)dx = 1; (4.16)

(ii) for any Borel set B of real numbers, we have

Mξ ∈ B =12

(supx∈B

λ(x) + supx∈<

λ(x)− supx∈Bc

λ(x))

+∫

B

ρ(x)dx. (4.17)

Remark 4.5: It is not true that all uncertain variables have their ownidentification functions.

Remark 4.6: The uncertain variable with identification function (λ, ρ) isessentially a fuzzy variable if

supx∈<

λ(x) = 1.

For this case, λ is a membership function and ρ ≡ 0, a.e.

Remark 4.7: The uncertain variable with identification function (λ, ρ) isessentially a random variable if∫

<ρ(x)dx = 1.

For this case, ρ is a probability density function and λ ≡ 0.

Remark 4.8: Let ξ be an uncertain variable with identification function(λ, ρ). If λ(x) is a continuous function, then we have

Mξ = x =λ(x)

2, ∀x ∈ <. (4.18)

Section 4.4 - Uncertainty Distribution 185

Theorem 4.8 Suppose λ(x) is a nonnegative function and ρ(x) is a nonneg-ative and integrable function satisfying (4.16). Then there is an uncertainvariable ξ such that (4.17) holds.

Proof: Let < be the universal set. For each Borel set B of real numbers, wedefine a set function

MB =12

(supx∈B

λ(x) + supx∈<

λ(x)− supx∈Bc

λ(x))

+∫

B

ρ(x)dx.

It is clear that M is normal, increasing, self-dual, and countably subadditive.That is, the set function M is indeed an uncertain measure. Now we definean uncertain variable ξ as an identity function from the uncertainty space(<,A,M) to <. We may verify that ξ meets (4.17). The theorem is proved.

4.4 Uncertainty Distribution

Definition 4.9 (Liu [132]) The uncertainty distribution Φ: < → [0, 1] of anuncertain variable ξ is defined by

Φ(x) = Mγ ∈ Γ

∣∣ ξ(γ) ≤ x. (4.19)

Theorem 4.9 An uncertainty distribution is an increasing function suchthat

0 ≤ limx→−∞

Φ(x) < 1, 0 < limx→+∞

Φ(x) ≤ 1. (4.20)

Proof: It is obvious that an uncertainty distribution Φ is an increasingfunction, and the inequalities (4.20) follow from the uncertainty asymptotictheorem immediately.

Definition 4.10 A continuous uncertain variable is said to be (a) singularif its uncertainty distribution is a singular function; (b) absolutely continuousif its uncertainty distribution is absolutely continuous.

Definition 4.11 (Liu [132]) The uncertainty density function φ: < → [0,+∞)of an uncertain variable ξ is a function such that

Φ(x) =∫ x

−∞φ(y)dy, ∀x ∈ <, (4.21)

∫ +∞

−∞φ(y)dy = 1 (4.22)

where Φ is the uncertainty distribution of ξ.


Example 4.8: The uncertainty density function may not exist even if theuncertainty distribution is continuous and differentiable a.e. Suppose f is theCantor function, and set

Φ(x) =

0, if x < 0

f(x), if 0 ≤ x ≤ 11, if x > 1.

(4.23)

Then Φ is an increasing and continuous function, and is an uncertainty dis-tribution. Note that Φ′(x) = 0 almost everywhere, and∫ +∞

−∞Φ′(x)dx = 0 6= 1.

Thus the uncertainty density function does not exist.

Theorem 4.10 Let ξ be an uncertain variable whose uncertainty densityfunction φ exists. Then we have

Mξ ≤ x =∫ x

−∞φ(y)dy, Mξ ≥ x =

∫ +∞

x

φ(y)dy. (4.24)

Proof: The first part follows immediately from the definition. In addition,by the self-duality of uncertain measure, we have

Mξ ≥ x = 1−Mξ < x =∫ +∞

−∞φ(y)dy −

∫ x

−∞φ(y)dy =

∫ +∞

x

φ(y)dy.


Joint Uncertainty Distribution

Definition 4.12 Let (ξ1, ξ2, · · · , ξn) be an uncertain vector. Then the jointuncertainty distribution Φ : <n → [0, 1] is defined by

Φ(x1, x2, · · · , xn) = Mγ ∈ Γ

∣∣ ξ1(γ) ≤ x1, ξ2(γ) ≤ x2, · · · , ξn(γ) ≤ xn

.

Definition 4.13 The joint uncertainty density function φ : <n → [0,+∞)of an uncertain vector (ξ1, ξ2, · · · , ξn) is a function such that

Φ(x1, x2, · · · , xn) =∫ x1

−∞

∫ x2

−∞· · ·∫ xn

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn


−∞

∫ +∞

−∞· · ·∫ +∞

−∞φ(y1, y2, · · · , yn)dy1dy2 · · ·dyn = 1

where Φ is the joint uncertainty distribution of (ξ1, ξ2, · · · , ξn).


4.5 Expected Value

Expected value is the average value of uncertain variable in the sense ofuncertain measure, and represents the size of uncertain variable.

Definition 4.14 (Liu [132]) Let ξ be an uncertain variable. Then the ex-pected value of ξ is defined by

E[ξ] =∫ +∞

0

Mξ ≥ rdr −∫ 0

−∞Mξ ≤ rdr (4.25)


Theorem 4.11 Let ξ be an uncertain variable whose uncertainty densityfunction φ exists. If the Lebesgue integral∫ +∞

−∞xφ(x)dx


E[ξ] =∫ +∞

−∞xφ(x)dx. (4.26)


E[ξ] =∫ +∞

0

Mξ ≥ rdr −∫ 0

−∞Mξ ≤ rdr

=∫ +∞

0

[∫ +∞

r

φ(x)dx]

dr −∫ 0

−∞

[∫ r

−∞φ(x)dx

]dr

=∫ +∞

0

[∫ x

0

φ(x)dr]

dx−∫ 0

−∞

[∫ 0

x

φ(x)dr]

dx

=∫ +∞

0

xφ(x)dx+∫ 0

−∞xφ(x)dx

=∫ +∞

−∞xφ(x)dx.


Theorem 4.12 Let ξ be an uncertain variable with uncertainty distributionΦ. If

limx→−∞

Φ(x) = 0, limx→∞

Φ(x) = 1



−∞xdΦ(x)


E[ξ] =∫ +∞

−∞xdΦ(x). (4.27)


diately have

limy→+∞

∫ y

0

xdΦ(x) =∫ +∞

0


∫ 0

y

xdΦ(x) =∫ 0

−∞xdΦ(x)

and

limy→+∞

∫ +∞

y


∫ y

−∞xdΦ(x) = 0.


y

xdΦ(x) ≥ y

(lim

z→+∞Φ(z)− Φ(y)

)= y (1− Φ(y)) ≥ 0, for y > 0,

∫ y

−∞xdΦ(x) ≤ y

(Φ(y)− lim

z→−∞Φ(z)

)= yΦ(y) ≤ 0, for y < 0

thatlim

y→+∞y (1− Φ(y)) = 0, lim

y→−∞yΦ(y) = 0.


n−1∑i=0

xi (Φ(xi+1)− Φ(xi)) →∫ y

0

xdΦ(x)

andn−1∑i=0

(1− Φ(xi+1))(xi+1 − xi) →∫ y

0

Mξ ≥ rdr


n−1∑i=0

xi (Φ(xi+1)− Φ(xi))−n−1∑i=0

(1− Φ(xi+1)(xi+1 − xi) = y(Φ(y)− 1) → 0


0

Mξ ≥ rdr =∫ +∞

0

xdΦ(x).



−∫ 0

−∞Mξ ≤ rdr =

∫ 0

−∞xdΦ(x).


Theorem 4.13 Let ξ be an uncertain variable with finite expected value.Then for any real numbers a and b, we have

E[aξ + b] = aE[ξ] + b. (4.28)


E[ξ + b] =∫ +∞

0

Mξ + b ≥ rdr −∫ 0

−∞Mξ + b ≤ rdr

=∫ +∞

0

Mξ ≥ r − bdr −∫ 0

−∞Mξ ≤ r − bdr

= E[ξ] +∫ b

0

(Mξ ≥ r − b+Mξ < r − b)dr

= E[ξ] + b.


E[aξ + b] = E[ξ]−∫ 0

b

(Mξ ≥ r − b+Mξ < r − b)dr = E[ξ] + b.

Step 2: We prove E[aξ] = aE[ξ]. If a = 0, then the equation E[aξ] =aE[ξ] holds trivially. If a > 0, we have

E[aξ] =∫ +∞

0

Maξ ≥ rdr −∫ 0

−∞Maξ ≤ rdr

=∫ +∞

0

Mξ ≥ r/adr −∫ 0

−∞Mξ ≤ r/adr

= a

∫ +∞

0

Mξ ≥ tdt− a

∫ 0

−∞Mξ ≤ tdt

= aE[ξ].

If a < 0, we have

E[aξ] =∫ +∞

0

Maξ ≥ rdr −∫ 0

−∞Maξ ≤ rdr

=∫ +∞

0

Mξ ≤ r/adr −∫ 0

−∞Mξ ≥ r/adr

= a

∫ +∞

0

Mξ ≥ tdt− a

∫ 0

−∞Mξ ≤ tdt

= aE[ξ].


Finally, for any real numbers a and b, it follows from Steps 1 and 2 that thetheorem holds.

Theorem 4.14 Let f be a convex function on [a, b], and ξ an uncertainvariable that takes values in [a, b] and has expected value e. Then

E[f(ξ)] ≤ b− e

b− af(a) +

e− a

b− af(b). (4.29)

Proof: For each γ ∈ Γ, we have a ≤ ξ(γ) ≤ b and

ξ(γ) =b− ξ(γ)b− a

a+ξ(γ)− a

b− ab.


f(ξ(γ)) ≤ b− ξ(γ)b− a

f(a) +ξ(γ)− a

b− af(b).

Taking expected values on both sides, we obtain the inequality.

4.6 Variance

The variance of an uncertain variable provides a measure of the spread of thedistribution around its expected value. A small value of variance indicatesthat the uncertain variable is tightly concentrated around its expected value;and a large value of variance indicates that the uncertain variable has a widespread around its expected value.

Definition 4.15 (Liu [132]) Let ξ be an uncertain variable with finite ex-pected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2].

Theorem 4.15 If ξ is an uncertain variable with finite expected value, a andb are real numbers, then V [aξ + b] = a2V [ξ].


V [aξ + b] = E[(aξ + b− aE[ξ]− b)2

]= a2E[(ξ − E[ξ])2] = a2V [ξ].

Theorem 4.16 Let ξ be an uncertain variable with expected value e. ThenV [ξ] = 0 if and only if Mξ = e = 1.


E[(ξ − e)2] =∫ +∞

0

M(ξ − e)2 ≥ rdr

which implies M(ξ − e)2 ≥ r = 0 for any r > 0. Hence we have

M(ξ − e)2 = 0 = 1.


That is, Mξ = e = 1.Conversely, if Mξ = e = 1, then we have M(ξ − e)2 = 0 = 1 and

M(ξ − e)2 ≥ r = 0 for any r > 0. Thus

V [ξ] =∫ +∞

0

M(ξ − e)2 ≥ rdr = 0.


Theorem 4.17 Let ξ be an uncertain variable that takes values in [a, b] andhas expected value e. Then

V [ξ] ≤ (e− a)(b− e). (4.30)

Proof: It follows from Theorem 4.14 immediately by defining f(x) = (x−e)2.

4.7 Moments

Definition 4.16 (Liu [132]) Let ξ be an uncertain variable. Then for anypositive integer k,(a) the expected value E[ξk] is called the kth moment;(b) the expected value E[|ξ|k] is called the kth absolute moment;(c) the expected value E[(ξ − E[ξ])k] is called the kth central moment;(d) the expected value E[|ξ−E[ξ]|k] is called the kth absolute central moment.


Theorem 4.18 Let ξ be a nonnegative uncertain variable, and k a positivenumber. Then the k-th moment

E[ξk] = k

∫ +∞

0

rk−1Mξ ≥ rdr. (4.31)


E[ξk] =∫ ∞

0

Mξk ≥ xdx =∫ ∞

0

Mξ ≥ rdrk = k

∫ ∞

0

rk−1Mξ ≥ rdr.


Theorem 4.19 Let ξ be an uncertain variable, and t a positive number. IfE[|ξ|t] <∞, then

limx→∞

xtM|ξ| ≥ x = 0. (4.32)

Conversely, if (4.32) holds for some positive number t, then E[|ξ|s] <∞ forany 0 ≤ s < t.


Proof: It follows from the definition of expected value operator that

E[|ξ|t] =∫ +∞

0

M|ξ|t ≥ rdr <∞.

Thus we have

limx→∞

∫ +∞

xt/2

M|ξ|t ≥ rdr = 0.

The equation (4.32) is proved by the following relation,∫ +∞

xt/2

M|ξ|t ≥ rdr ≥∫ xt

xt/2

M|ξ|t ≥ rdr ≥ 12xtM|ξ| ≥ x.

Conversely, if (4.32) holds, then there exists a number a such that

xtM|ξ| ≥ x ≤ 1, ∀x ≥ a.

Thus we have

E[|ξ|s] =∫ a

0

M|ξ|s ≥ rdr +∫ +∞

a

M|ξ|s ≥ rdr

=∫ a

0

M|ξ|s ≥ rdr +∫ +∞

a

srs−1M|ξ| ≥ rdr

≤∫ a

0

M|ξ|s ≥ rdr + s

∫ +∞

0

rs−t−1dr

< +∞.

(by∫ +∞

0

rpdr <∞ for any p < −1)


Theorem 4.20 Let ξ be an uncertain variable that takes values in [a, b] andhas expected value e. Then for any positive integer k, the kth absolute momentand kth absolute central moment satisfy the following inequalities,

E[|ξ|k] ≤ b− e

b− a|a|k +

e− a

b− a|b|k, (4.33)

E[|ξ − e|k] ≤ b− e

b− a(e− a)k +

e− a

b− a(b− e)k. (4.34)



4.8 Independence

Definition 4.17 (Liu [132]) The uncertain variables ξ1, ξ2, · · · , ξn are saidto be independent if

E

[n∑

i=1

fi(ξi)

]=

n∑i=1

E[fi(ξi)] (4.35)

for any measurable functions f1, f2, · · · , fn provided that the expected valuesexist and are finite.

What are the sufficient conditions for independence of uncertain variables?This is an open problem.

Theorem 4.21 If ξ and η are independent uncertain variables with finiteexpected values, then we have

E[aξ + bη] = aE[ξ] + bE[η] (4.36)

for any real numbers a and b.

Proof: The theorem follows from the definition by defining f1(x) = ax andf2(x) = bx.

Theorem 4.22 Suppose that ξ1, ξ2, · · · , ξn are independent uncertain vari-ables, and f1, f2, · · · , fn are measurable functions. Then the uncertain vari-ables f1(ξ1), f2(ξ2), · · · , fn(ξn) are independent.

Proof: The theorem follows from the definition because the compound ofmeasurable functions is also measurable.


This section introduces the concept of identical distribution of uncertain vari-ables.

Definition 4.18 The uncertain variables ξ and η are identically distributedif

Mξ ∈ B = Mη ∈ B (4.37)

for any Borel set B of real numbers.

Theorem 4.23 Let ξ and η be identically distributed uncertain variables,and f : < → < a measurable function. Then f(ξ) and f(η) are identicallydistributed uncertain variables.


Proof: For any Borel set B of real numbers, we have

Mf(ξ) ∈ B = Mξ ∈ f−1(B) = Mη ∈ f−1(B) = Mf(η) ∈ B.

Hence f(ξ) and f(η) are identically distributed uncertain variables.

Theorem 4.24 If ξ and η are identically distributed uncertain variables,then they have the same uncertainty distribution.

Proof: Since ξ and η are identically distributed uncertain variables, we haveMξ ∈ (−∞, x] = Mη ∈ (−∞, x] for any x. Thus ξ and η have the sameuncertainty distribution.

Theorem 4.25 If ξ and η are identically distributed uncertain variableswhose uncertainty density functions exist, then they have the same uncer-tainty density function.

Proof: It follows from Theorem 4.24 immediately.

4.10 Critical Values

Definition 4.19 (Liu [132]) Let ξ be an uncertain variable, and α ∈ (0, 1].Then

ξsup(α) = supr∣∣M ξ ≥ r ≥ α

(4.38)


ξinf(α) = infr∣∣M ξ ≤ r ≥ α

(4.39)


Theorem 4.26 Let ξ be an uncertain variable and α a number between 0and 1. We have(a) if c ≥ 0, then (cξ)sup(α) = cξsup(α) and (cξ)inf(α) = cξinf(α);(b) if c < 0, then (cξ)sup(α) = cξinf(α) and (cξ)inf(α) = cξsup(α).

Proof: (a) If c = 0, then the part (a) is obvious. In the case of c > 0, wehave

(cξ)sup(α) = supr∣∣Mcξ ≥ r ≥ α

= c supr/c | Mξ ≥ r/c ≥ α

= cξsup(α).

A similar way may prove (cξ)inf(α) = cξinf(α). In order to prove the part (b),it suffices to prove that (−ξ)sup(α) = −ξinf(α) and (−ξ)inf(α) = −ξsup(α).In fact, we have

(−ξ)sup(α) = supr∣∣M−ξ ≥ r ≥ α

= − inf−r | Mξ ≤ −r ≥ α

= −ξinf(α).



Theorem 4.27 Let ξ be an uncertain variable. Then we have(a) if α > 0.5, then ξinf(α) ≥ ξsup(α);(b) if α ≤ 0.5, then ξinf(α) ≤ ξsup(α).


1 ≥Mξ < ξ(α)+Mξ > ξ(α) ≥ α+ α > 1.

A contradiction proves ξinf(α) ≥ ξsup(α).Part (b): Assume that ξinf(α) > ξsup(α). It follows from the definition

of ξinf(α) that Mξ ≤ ξ(α) < α. Similarly, it follows from the definition ofξsup(α) that Mξ ≥ ξ(α) < α. Thus

1 ≤Mξ ≤ ξ(α)+Mξ ≥ ξ(α) < α+ α ≤ 1.

A contradiction proves ξinf(α) ≤ ξsup(α). The theorem is verified.

Theorem 4.28 Let ξ be an uncertain variable. Then ξsup(α) is a decreasingfunction of α, and ξinf(α) is an increasing function of α.

Proof: It follows from the definition immediately.

4.11 Entropy

This section provides a definition of entropy to characterize the uncertaintyof uncertain variables resulting from information deficiency.

Definition 4.20 (Liu [132]) Suppose that ξ is a discrete uncertain variabletaking values in x1, x2, · · · . Then its entropy is defined by

H[ξ] =∞∑

i=1

S(Mξ = xi) (4.40)


Example 4.9: Suppose that ξ is a discrete uncertain variable taking valuesin x1, x2, · · · . If there exists some index k such that Mξ = xk = 1, and0 otherwise, then its entropy H[ξ] = 0.

Example 4.10: Suppose that ξ is a simple uncertain variable taking valuesin x1, x2, · · · , xn. If Mξ = xi = 0.5 for all i = 1, 2, · · · , n, then itsentropy H[ξ] = n ln 2.

Theorem 4.29 Suppose that ξ is a discrete uncertain variable taking valuesin x1, x2, · · · . Then

H[ξ] ≥ 0 (4.41)

and equality holds if and only if ξ is essentially a deterministic/crisp number.


Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only ifMξ = xi = 0 or 1 for each i. That is, there exists one and only one index ksuch that Mξ = xk = 1, i.e., ξ is essentially a deterministic/crisp number.

This theorem states that the entropy of an uncertain variable reaches itsminimum 0 when the uncertain variable degenerates to a deterministic/crispnumber. In this case, there is no uncertainty.

Theorem 4.30 Suppose that ξ is a simple uncertain variable taking valuesin x1, x2, · · · , xn. Then

H[ξ] ≤ n ln 2 (4.42)

and equality holds if and only if Mξ = xi = 0.5 for all i = 1, 2, · · · , n.


H[ξ] =n∑

i=1

S(Mξ = xi) ≤ n ln 2

and equality holds if and only if Mξ = xi = 0.5 for all i = 1, 2, · · · , n.

This theorem states that the entropy of an uncertain variable reaches itsmaximum when the uncertain variable is an equipossible one. In this case,there is no preference among all the values that the uncertain variable willtake.

4.12 Distance

Definition 4.21 (Liu [132]) The distance between uncertain variables ξ andη is defined as

d(ξ, η) = E[|ξ − η|]. (4.43)

Theorem 4.31 Let ξ, η, τ be uncertain variables, and let d(·, ·) be the dis-tance. Then we have(a) (Nonnegativity) d(ξ, η) ≥ 0;(b) (Identification) d(ξ, η) = 0 if and only if ξ = η;(c) (Symmetry) d(ξ, η) = d(η, ξ);(d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ) + 2d(η, τ).

Proof: The parts (a), (b) and (c) follow immediately from the definition.Now we prove the part (d). It follows from the countable subadditivity axiom


that

d(ξ, η) =∫ +∞

0

M |ξ − η| ≥ rdr

≤∫ +∞

0

M |ξ − τ |+ |τ − η| ≥ rdr

≤∫ +∞

0

M |ξ − τ | ≥ r/2 ∪ |τ − η| ≥ r/2dr

≤∫ +∞

0

(M|ξ − τ | ≥ r/2+M|τ − η| ≥ r/2) dr

=∫ +∞

0

M|ξ − τ | ≥ r/2dr +∫ +∞

0

M|τ − η| ≥ r/2dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ) + 2d(τ, η).

Example 4.11: Let Γ = γ1, γ2, γ3. Define M∅ = 0, MΓ = 1 andMΛ = 1/2 for any subset Λ (excluding ∅ and Γ). We set uncertain variablesξ, η and τ as follows,

ξ(γ) =

1, if γ 6= γ3

0, otherwise,η(γ) =

−1, if γ 6= γ1

0, otherwise,τ(γ) ≡ 0.

It is easy to verify that d(ξ, τ) = d(τ, η) = 1/2 and d(ξ, η) = 3/2. Thus

d(ξ, η) =32(d(ξ, τ) + d(τ, η)).

4.13 Inequalities

Theorem 4.32 (Liu [132]) Let ξ be an uncertain variable, and f a non-negative function. If f is even and increasing on [0,∞), then for any givennumber t > 0, we have

M|ξ| ≥ t ≤ E[f(ξ)]f(t)

. (4.44)


Proof: It is clear that M|ξ| ≥ f−1(r) is a monotone decreasing functionof r on [0,∞). It follows from the nonnegativity of f(ξ) that

E[f(ξ)] =∫ +∞

0

Mf(ξ) ≥ rdr

=∫ +∞

0

M|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

M|ξ| ≥ f−1(r)dr

≥∫ f(t)

0

dr ·M|ξ| ≥ f−1(f(t))

= f(t) ·M|ξ| ≥ t


Theorem 4.33 (Liu [132], Markov Inequality) Let ξ be an uncertain vari-able. Then for any given numbers t > 0 and p > 0, we have

M|ξ| ≥ t ≤ E[|ξ|p]tp

. (4.45)


Theorem 4.34 (Liu [132], Chebyshev Inequality) Let ξ be an uncertain vari-able whose variance V [ξ] exists. Then for any given number t > 0, we have

M |ξ − E[ξ]| ≥ t ≤ V [ξ]t2

. (4.46)

Proof: It is a special case of Theorem 4.32 when the uncertain variable ξ isreplaced with ξ − E[ξ], and f(x) = x2.

Theorem 4.35 (Liu [132], Holder’s Inequality) Let p and q be positive realnumbers with 1/p+ 1/q = 1, and let ξ and η be independent uncertain vari-ables with E[|ξ|p] <∞ and E[|η|q] <∞. Then we have

E[|ξη|] ≤ p√E[|ξ|p] q

√E[|η|q]. (4.47)









E[f(|ξ|p, |η|q)] ≤ f(E[|ξ|p], E[|η|q]).


Theorem 4.36 (Liu [132], Minkowski Inequality) Let p be a real numberwith p ≥ 1, and let ξ and η be independent uncertain variables with E[|ξ|p] <∞ and E[|η|p] <∞. Then we have

p√E[|ξ + η|p] ≤ p

√E[|ξ|p] + p

√E[|η|p]. (4.48)


√x + p







E[f(|ξ|p, |η|p)] ≤ f(E[|ξ|p], E[|η|p]).


Theorem 4.37 (Liu [132], Jensen’s Inequality) Let ξ be an uncertain vari-able, and f : < → < a convex function. If E[ξ] and E[f(ξ)] are finite, then

f(E[ξ]) ≤ E[f(ξ)]. (4.49)



f(ξ)− f(E[ξ]) ≥ k · (ξ − E[ξ]).


E[f(ξ)]− f(E[ξ]) ≥ k · (E[ξ]− E[ξ]) = 0




We have the following four convergence concepts of uncertain sequence: con-vergence almost surely (a.s.), convergence in measure, convergence in mean,and convergence in distribution.

Table 4.1: Relationship among Convergence Concepts

Convergence⇒

Convergence⇒

Convergence

in Mean in Measure in Distribution

Definition 4.22 (Liu [132]) Suppose that ξ, ξ1, ξ2, · · · are uncertain vari-ables defined on the uncertainty space (Γ,L,M). The sequence ξi is saidto be convergent a.s. to ξ if there exists an event Λ with MΛ = 1 such that

limi→∞

|ξi(γ)− ξ(γ)| = 0 (4.50)

for every γ ∈ Λ. In that case we write ξi → ξ, a.s.

Definition 4.23 (Liu [132]) Suppose that ξ, ξ1, ξ2, · · · are uncertain vari-ables. We say that the sequence ξi converges in measure to ξ if

limi→∞

M |ξi − ξ| ≥ ε = 0 (4.51)

for every ε > 0.

Definition 4.24 (Liu [132]) Suppose that ξ, ξ1, ξ2, · · · are uncertain vari-ables with finite expected values. We say that the sequence ξi converges inmean to ξ if

limi→∞

E[|ξi − ξ|] = 0. (4.52)


limi→∞

E[|ξi − ξ|2] = 0. (4.53)

Definition 4.25 (Liu [132]) Suppose that Φ,Φ1,Φ2, · · · are the uncertaintydistributions of uncertain variables ξ, ξ1, ξ2, · · · , respectively. We say thatξi converges in distribution to ξ if Φi → Φ at any continuity point of Φ.

Theorem 4.26 (Liu [132]) Suppose that ξ, ξ1, ξ2, · · · are uncertain variables.If ξi converges in mean to ξ, then ξi converges in measure to ξ.

Proof: It follows from the Markov inequality that for any given numberε > 0, we have

M|ξi − ξ| ≥ ε ≤ E[|ξi − ξ|]ε

→ 0

as i→∞. Thus ξi converges in measure to ξ. The theorem is proved.

Section 4.15 - Conditional Uncertainty 201

Theorem 4.27 (Liu [132]) Suppose ξ, ξ1, ξ2, · · · are uncertain variables. Ifξi converges in measure to ξ, then ξi converges in distribution to ξ.

Proof: Let x be a given continuity point of the uncertainty distribution Φ.On the one hand, for any y > x, we have


It follows from the countable subadditivity axiom that

Φi(x) ≤ Φ(y) +M|ξi − ξ| ≥ y − x.

Since ξi converges in measure to ξ, we have M|ξi − ξ| ≥ y − x → 0 asi → ∞. Thus we obtain lim supi→∞ Φi(x) ≤ Φ(y) for any y > x. Lettingy → x, we get

lim supi→∞

Φi(x) ≤ Φ(x). (4.54)


ξ ≤ z = ξi ≤ x, ξ ≤ z ∪ ξi > x, ξ ≤ z ⊂ ξi ≤ x ∪ |ξi − ξ| ≥ x− z

which implies that

Φ(z) ≤ Φi(x) +M|ξi − ξ| ≥ x− z.

Since M|ξi − ξ| ≥ x − z → 0, we obtain Φ(z) ≤ lim infi→∞ Φi(x) for anyz < x. Letting z → x, we get


Φi(x). (4.55)


4.15 Conditional Uncertainty

We consider the uncertain measure of an event A after it has been learnedthat some other event B has occurred. This new uncertain measure of A iscalled the conditional uncertain measure of A given B.

In order to define a conditional uncertain measure MA|B, at first wehave to enlarge MA ∩ B because MA ∩ B < 1 for all events wheneverMB < 1. It seems that we have no alternative but to divide MA∩B byMB. Unfortunately, MA∩B/MB is not always an uncertain measure.However, the value MA|B should not be greater than MA ∩ B/MB(otherwise the normality will be lost), i.e.,

MA|B ≤ MA ∩BMB

. (4.56)



MA|B = 1−MAc|B ≥ 1− MAc ∩BMB

. (4.57)

Furthermore, since (A ∩B) ∪ (Ac ∩B) = B, we have MB ≤MA ∩B+MAc ∩B by using the countable subadditivity axiom. Thus

0 ≤ 1− MAc ∩BMB

≤ MA ∩BMB

≤ 1. (4.58)

Hence any numbers between 1−MAc∩B/MB andMA∩B/MB arereasonable values that the conditional uncertain measure may take. Basedon the maximum uncertainty principle, we have the following conditionaluncertain measure.

Definition 4.28 (Liu [132]) Let (Γ,L,M) be an uncertainty space, and A,B ∈L. Then the conditional uncertain measure of A given B is defined by

MA|B =

MA ∩BMB

, ifMA ∩BMB

< 0.5

1− MAc ∩BMB

, ifMAc ∩BMB

< 0.5

0.5, otherwise

(4.59)

provided that MB > 0.

It follows immediately from the definition of conditional uncertain mea-sure that

1− MAc ∩BMB

≤MA|B ≤ MA ∩BMB

. (4.60)

Furthermore, the conditional uncertain measure obeys the maximum uncer-tainty principle, and takes values as close to 0.5 as possible.

Remark 4.9: Conditional uncertain measure coincides with conditionalprobability, conditional credibility, and conditional chance.

Theorem 4.38 (Liu [132]) Let (Γ,L,M) be an uncertainty space, and Ban event with MB > 0. Then M·|B defined by (4.59) is an uncertainmeasure, and (Γ,L,M·|B) is an uncertainty space.

Proof: It is sufficient to prove that M·|B satisfies the normality, mono-tonicity, self-duality and countable subadditivity axioms. At first, it satisfiesthe normality axiom, i.e.,

MΓ|B = 1− MΓc ∩BMB

= 1− M∅MB

= 1.

Section 4.15 - Conditional Uncertainty 203


MA1 ∩BMB

≤ MA2 ∩BMB

< 0.5,

then

MA1|B =MA1 ∩BMB

≤ MA2 ∩BMB

= MA2|B.

IfMA1 ∩BMB

≤ 0.5 ≤ MA2 ∩BMB

,

then MA1|B ≤ 0.5 ≤MA2|B. If

0.5 <MA1 ∩BMB

≤ MA2 ∩BMB

,

then we have

MA1|B =(1− MAc

1 ∩BMB

)∨0.5 ≤

(1− MAc

2 ∩BMB

)∨0.5 = MA2|B.

This means that M·|B satisfies the monotonicity axiom. For any event A,if

MA ∩BMB

≥ 0.5,MAc ∩BMB

≥ 0.5,

then we have MA|B+MAc|B = 0.5 + 0.5 = 1 immediately. Otherwise,without loss of generality, suppose

MA ∩BMB

< 0.5 <MAc ∩BMB

,

then we have

MA|B+MAc|B =MA ∩BMB

+(

1− MA ∩BMB

)= 1.

That is, M·|B satisfies the self-duality axiom. Finally, for any countablesequence Ai of events, if MAi|B < 0.5 for all i, it follows from thecountable subadditivity axiom that

M

∞⋃i=1

Ai ∩B

≤M

∞⋃i=1

Ai ∩B

MB

≤

∞∑i=1

MAi ∩B

MB=

∞∑i=1

MAi|B.

Suppose there is one term greater than 0.5, say

MA1|B ≥ 0.5, MAi|B < 0.5, i = 2, 3, · · ·


If M∪iAi|B = 0.5, then we immediately have

M

∞⋃i=1

Ai ∩B

≤

∞∑i=1

MAi|B.

If M∪iAi|B > 0.5, we may prove the above inequality by the followingfacts:

Ac1 ∩B ⊂

∞⋃i=2

(Ai ∩B) ∪

( ∞⋂i=1

Aci ∩B

),

MAc1 ∩B ≤

∞∑i=2

MAi ∩B+M

∞⋂i=1

Aci ∩B

,

M

∞⋃i=1

Ai|B

= 1−

M

∞⋂i=1

Aci ∩B

MB

,

∞∑i=1

MAi|B ≥ 1− MAc1 ∩B

MB+

∞∑i=2

MAi ∩B

MB.

If there are at least two terms greater than 0.5, then the countable subad-ditivity is clearly true. Thus M·|B satisfies the countable subadditivityaxiom. Hence M·|B is an uncertain measure. Furthermore, (Γ,L,M·|B)is an uncertainty space.

Example 4.12: Let ξ and η be two uncertain variables. Then we have

M ξ = x|η = y =

Mξ = x, η = yMη = y

, ifMξ = x, η = yMη = y

< 0.5

1− Mξ 6= x, η = yMη = y

, ifMξ 6= x, η = yMη = y

< 0.5

0.5, otherwise

provided that Mη = y > 0.

Definition 4.29 (Liu [132]) The conditional uncertainty distribution Φ: < →[0, 1] of an uncertain variable ξ given B is defined by

Φ(x|B) = M ξ ≤ x|B (4.61)

provided that MB > 0.

Section 4.16 - Uncertain Process 205

Example 4.13: Let ξ and η be uncertain variables. Then the conditionaluncertainty distribution of ξ given η = y is

Φ(x|η = y) =

Mξ ≤ x, η = yMη = y

, ifMξ ≤ x, η = yMη = y

< 0.5

1− Mξ > x, η = yMη = y

, ifMξ > x, η = yMη = y

< 0.5

0.5, otherwise

provided that Mη = y > 0.

Definition 4.30 (Liu [132]) The conditional uncertainty density function φof an uncertain variable ξ given B is a nonnegative function such that

Φ(x|B) =∫ x

−∞φ(y|B)dy, ∀x ∈ <, (4.62)

∫ +∞

−∞φ(y|B)dy = 1 (4.63)

where Φ(x|B) is the conditional uncertainty distribution of ξ given B.

Definition 4.31 (Liu [132]) Let ξ be an uncertain variable. Then the con-ditional expected value of ξ given B is defined by

E[ξ|B] =∫ +∞

0

Mξ ≥ r|Bdr −∫ 0

−∞Mξ ≤ r|Bdr (4.64)


Following conditional uncertain measure and conditional expected value,we also have conditional variance, conditional moments, conditional criticalvalues, conditional entropy as well as conditional convergence.

4.16 Uncertain Process

Definition 4.32 (Liu [133]) Let T be an index set and let (Γ,L,M) be anuncertainty space. An uncertain process is a measurable function from T ×(Γ,L,M) to the set of real numbers, i.e., for each t ∈ T and any Borel set Bof real numbers, the set

γ ∈ Γ∣∣ X(t, γ) ∈ B (4.65)

is an event.


That is, an uncertain process Xt(γ) is a function of two variables suchthat the function Xt∗(γ) is an uncertain variable for each t∗. For each fixedγ∗, the function Xt(γ∗) is called a sample path of the uncertain process. Anuncertain process Xt(γ) is said to be sample-continuous if the sample pathis continuous for almost all γ.

Definition 4.33 (Liu [133]) An uncertain process Xt is said to have inde-pendent increments if

Xt1 −Xt0 , Xt2 −Xt1 , · · · , Xtk−Xtk−1 (4.66)

are independent uncertain variables for any times t0 < t1 < · · · < tk. Anuncertain process Xt is said to have stationary increments if, for any givent > 0, the increments Xs+t−Xs are identically distributed uncertain variablesfor all s > 0.

Uncertain Renewal Process

Definition 4.34 (Liu [133]) Let ξ1, ξ2, · · · be iid positive uncertain variables.Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn for n ≥ 1. Then the uncertainprocess

Nt = maxn≥0

n∣∣ Sn ≤ t

(4.67)

is called an uncertain renewal process.


can be regarded as the waiting time until the occurrence of the nth event,and Nt is the number of renewals in (0, t]. Each sample path of Nt is aright-continuous and increasing step function taking only nonnegative integervalues. Furthermore, the size of each jump of Nt is always 1. In other words,Nt has at most one renewal at each time. In particular, Nt does not jump attime 0. Since Nt ≥ n is equivalent to Sn ≤ t, we immediately have

MNt ≥ n = MSn ≤ t. (4.68)

Theorem 4.39 (Liu [133]) Let Nt be an uncertain renewal process. Thenwe have

E[Nt] =∞∑

n=1

MSn ≤ t. (4.69)


E[Nt] =∫ ∞

0

MNt ≥ rdr =∞∑

n=1

∫ n

n−1

MNt ≥ rdr

=∞∑

n=1

MNt ≥ n =∞∑

n=1

MSn ≤ t.


Section 4.16 - Uncertain Process 207

Canonical Process

Definition 4.35 (Liu [133]) An uncertain process Wt is said to be a canon-ical process if(i) W0 = 0 and Wt is sample-continuous,(ii) Wt has stationary and independent increments,(iii) W1 is an uncertain variable with expected value 0 and variance 1.

Theorem 4.40 (Existence Theorem) There is a canonical process.

Proof: In fact, standard Brownian motion and standard C process are in-stances of canonical process.

Example 4.14: Let Bt be a standard Brownian motion, and Ct a standardC process. Then for each number a ∈ [0, 1], we may verify that

Wt = aBt + (1− a)Ct (4.70)

is a canonical process.

Theorem 4.41 Let Wt be a canonical process. Then E[Wt] = 0 for any t.

Proof: Let f(t) = E[Wt]. Then for any times t1 and t2, by using theproperty of stationary and independent increments, we obtain

f(t1 + t2) = E[Wt1+t2 ] = E[Wt1+t2 −Wt2 +Wt2 −W0]

= E[Wt1 ] + E[Wt2 ] = f(t1) + f(t2)

which implies that there is a constant e such that f(t) = et. The theorem isproved via f(1) = 0.

Theorem 4.42 Let Wt be a canonical process. If for any t1 and t2, we have

V [Wt1+t2 ] = V [Wt1 ] + V [Wt2 ],

then V [Wt] = t for any t.

Proof: Let f(t) = V [Wt]. Then for any times t1 and t2, by using the propertyof stationary and independent increments as well as variance condition, weobtain

f(t1 + t2) = V [Wt1+t2 ] = V [Wt1+t2 −Wt2 +Wt2 −W0]

= V [Wt1 ] + V [Wt2 ] = f(t1) + f(t2)

which implies that there is a constant σ such that f(t) = σ2t. It follows fromf(1) = 1 that σ2 = 1 and f(t) = t. Hence the theorem is proved.


Theorem 4.43 Let Wt be a canonical process. If for any t1 and t2, we have√V [Wt1+t2 ] =

√V [Wt1 ] +

√V [Wt2 ],

then V [Wt] = t2 for any t.

Proof: Let f(t) =√V [Wt]. Then for any times t1 and t2, by using the prop-

erty of stationary and independent increments as well as variance condition,we obtain

f(t1 + t2) =√V [Wt1+t2 ] =

√V [Wt1+t2 −Wt2 +Wt2 −W0]

=√V [Wt1 ] +

√V [Wt2 ] = f(t1) + f(t2)

which implies that there is a constant σ such that f(t) = σt. It follows fromf(1) = 1 that σ = 1 and f(t) = t. The theorem is verified.

Definition 4.36 For any partition of closed interval [0, t] with 0 = t1 < t2 <· · · < tk+1 = t, the mesh is written as

∆ = max1≤i≤k

|ti+1 − ti|.

Let α > 0 be a real number. Then the α-variation of uncertain process Wt is

lim∆→0

k∑i=1

|Wti −Wti |α (4.71)

provided that the limit exists in mean square and is an uncertain process.Especially, the α-variation is called total variation if α = 1; and squaredvariation if α = 2.

Definition 4.37 (Liu [133]) Let Wt be a canonical process. Then et+ σWt

is called a derived canonical process, and the uncertain process

Gt = exp(et+ σWt) (4.72)

is called a geometric canonical process.

An Uncertain Stock Model

Assume that stock price follows geometric canonical process. Then we havean uncertain stock model in which the bond price Xt and the stock price Yt

are determined by Xt = X0 exp(rt)

Yt = Y0 exp(et+ σWt)(4.73)

where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion,and Wt is a canonical process.

Section 4.17 - Uncertain Calculus 209

4.17 Uncertain Calculus

Let Wt be a canonical process, and dt an infinitesimal time interval. Then

dWt = Wt+dt −Wt (4.74)

is an uncertain process with E[dWt] = 0 and dt2 ≤ E[dW 2t ] ≤ dt.

Definition 4.38 (Liu [133]) Let Xt be an uncertain process and let Wt bea canonical process. For any partition of closed interval [a, b] with a = t1 <t2 < · · · < tk+1 = b, the mesh is written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the uncertain integral of Xt with respect to Wt is

∫ b

a

XtdWt = lim∆→0

k∑i=1

Xti· (Wti+1 −Wti

) (4.75)

provided that the limit exists in mean square and is an uncertain variable.

Example 4.15: Let Wt be a canonical process. Then for any partition0 = t1 < t2 < · · · < tk+1 = s, we have

∫ s

0

dWt = lim∆→0

k∑i=1

(Wti+1 −Wti) ≡Ws −W0 = Ws.

Example 4.16: Let Wt be a canonical process. Then for any partition0 = t1 < t2 < · · · < tk+1 = s, we have

sWs =k∑

i=1

(ti+1Wti+1 − tiWti

)=

k∑i=1

ti(Wti+1 −Wti) +k∑

i=1

Wti+1(ti+1 − ti)

→∫ s

0

tdWt +∫ s

0

Wtdt


0

tdWt = sWs −∫ s

0

Wtdt.


Theorem 4.44 (Liu [133]) Let Wt be a canonical process, and let h(t, w) bea twice continuously differentiable function. Define Xt = h(t,Wt). Then wehave the following chain rule

dXt =∂h

∂t(t,Wt)dt+

∂h

∂w(t,Wt)dWt +

12∂2h

∂w2(t,Wt)dW 2

t . (4.76)


∆Xt =∂h

∂t(t,Wt)∆t+

∂h

∂w(t,Wt)∆Wt +

12∂2h

∂w2(t,Wt)(∆Wt)2

+12∂2h

∂t2(t,Wt)(∆t)2 +

∂2h

∂t∂w(t,Wt)∆t∆Wt.

Since we can ignore the terms (∆t)2 and ∆t∆Wt, the chain rule is provedbecause it makes

Xs = X0 +∫ s

0

∂h

∂t(t,Wt)dt+

∫ s

0

∂h

∂w(t,Wt)dWt +

12

∫ s

0

∂2h

∂w2(t,Wt)dW 2

t

for any s ≥ 0.

Remark 4.10: The infinitesimal increment dWt in (4.76) may be replacedwith the derived canonical process

dYt = utdt+ vtdWt (4.77)

where ut is an absolutely integrable uncertain process, and vt is a squareintegrable uncertain process, thus producing

dh(t, Yt) =∂h

∂t(t, Yt)dt+

∂h

∂w(t, Yt)dYt +

12∂2h

∂w2(t, Yt)v2

t dW 2t . (4.78)

Example 4.17: Applying the chain rule, we obtain the following formula

d(tWt) = Wtdt+ tdWt.

Hence we have

sWs =∫ s

0

d(tWt) =∫ s

0

Wtdt+∫ s

0

tdWt.

That is, ∫ s

0

tdWt = sWs −∫ s

0

Wtdt.

Section 4.18 - Uncertain Differential Equation 211

Theorem 4.45 (Liu [133], Integration by Parts) Suppose that Wt is a canon-ical process and F (t) is an absolutely continuous function. Then∫ s

0

F (t)dWt = F (s)Ws −∫ s

0

WtdF (t). (4.79)

Proof: By defining h(t,Wt) = F (t)Wt and using the chain rule, we get

d(F (t)Wt) = WtdF (t) + F (t)dWt.

ThusF (s)Ws =

∫ s

0

d(F (t)Wt) =∫ s

0

WtdF (t) +∫ s

0

F (t)dWt


4.18 Uncertain Differential Equation

Definition 4.39 (Liu [133]) Suppose Wt is a canonical process, and f andg are some given functions. Then

dXt = f(t,Xt)dt+ g(t,Xt)dWt (4.80)

is called an uncertain differential equation. A solution is an uncertain processXt that satisfies (4.80) identically in t.

Remark 4.11: Note that there is no precise definition for the terms dXt, dtand dWt in the uncertain differential equation (4.80). The mathematicallymeaningful form is the uncertain integral equation

Xs = X0 +∫ s

0

f(t,Xt)dt+∫ s

0

g(t,Xt)dWt. (4.81)


Example 4.18: Let Wt be a canonical process. Then the uncertain differ-ential equation

dXt = adt+ bdWt

has a solution Xt = at+ bWt.

Appendix A

Measurable Sets

Algebra and σ-algebra are very important and fundamental concepts in mea-sure theory.

Definition A.1 Let Ω be a nonempty set. A collection A is called an algebraover Ω if the following conditions hold:(a) Ω ∈ A;(b) if A ∈ A, then Ac ∈ A;(c) if Ai ∈ A for i = 1, 2, · · · , n, then ∪n

i=1Ai ∈ A.If the condition (c) is replaced with closure under countable union, then A iscalled a σ-algebra over Ω.

Example A.1: Assume that Ω is a nonempty set. Then ∅,Ω is thesmallest σ-algebra over Ω, and the power set P (all subsets of Ω) is thelargest σ-algebra over Ω.

Example A.2: Let A be the set of all finite disjoint unions of all intervalsof the form (−∞, a], (a, b], (b,∞) and ∅. Then A is an algebra over <, butnot a σ-algebra because Ai = (0, (i− 1)/i] ∈ A for all i but

∞⋃i=1

Ai = (0, 1) 6∈ A.

Theorem A.1 A σ-algebra A is closed under difference, countable union,countable intersection, upper limit, lower limit, and limit. That is,

A2 \A1 ∈ A;∞⋃

i=1

Ai ∈ A;∞⋂

i=1

Ai ∈ A; (A.1)

lim supi→∞

Ai =∞⋂

k=1

∞⋃i=k

Ai ∈ A; (A.2)

214 Appendix A - Measurable Sets

lim infi→∞

Ai =∞⋃

k=1

∞⋂i=k

Ai ∈ A; (A.3)

limi→∞

Ai ∈ A. (A.4)

Theorem A.2 The intersection of any collection of σ-algebras is a σ-algebra.Furthermore, for any nonempty class C, there is a unique minimal σ-algebracontaining C.

Theorem A.3 (Monotone Class Theorem) Assume that A0 is an algebraover Ω, and C is a monotone class of subsets of Ω (if Ai ∈ C and Ai ↑ Aor Ai ↓ A, then A ∈ C). If A0 ⊂ C and σ(A0) is the smallest σ-algebracontaining A0, then σ(A0) ⊂ C.

Definition A.2 Let Ω be a nonempty set, and A a σ-algebra over Ω. Then(Ω,A) is called a measurable space, and any elements in A are called mea-surable sets.

Theorem A.4 The smallest σ-algebra B containing all open intervals iscalled the Borel algebra of <, any elements in B are called a Borel set.

Borel algebra B is a special σ-algebra over <. It follows from Theorem A.2that Borel algebra is unique. It has been proved that open set, closed set,the set of rational numbers, the set of irrational numbers, and countable setof real numbers are all Borel sets.

Example A.3: We divide the interval [0, 1] into three equal open intervalsfrom which we choose the middle one, i.e., (1/3, 2/3). Then we divide eachof the remaining two intervals into three equal open intervals, and choosethe middle one in each case, i.e., (1/9, 2/9) and (7/9, 8/9). We perform thisprocess and obtain Dij for j = 1, 2, · · · , 2i−1 and i = 1, 2, · · · Note that Dijis a sequence of mutually disjoint open intervals. Without loss of generality,suppose Di1 < Di2 < · · · < Di,2i−1 for i = 1, 2, · · · Define the set

D =∞⋃

i=1

2i−1⋃j=1

Dij . (A.5)

Then C = [0, 1] \ D is called the Cantor set. In other words, x ∈ C if andonly if x can be expressed in ternary form using only digits 0 and 2, i.e.,

x =∞∑

i=1

ai

3i(A.6)

where ai = 0 or 2 for i = 1, 2, · · · The Cantor set is closed, uncountable, anda Borel set.

Let Ω1,Ω2, · · · ,Ωn be any sets (not necessarily subsets of the same space).The product Ω = Ω1 ×Ω2 × · · · ×Ωn is the set of all ordered n-tuples of theform (x1, x2, · · · , xn), where xi ∈ Ωi for i = 1, 2, · · · , n.

Appendix A - Measurable Sets 215

Definition A.3 Let Ai be σ-algebras over Ωi, i = 1, 2, · · · , n, respectively.Write Ω = Ω1 × Ω2 × · · · × Ωn. A measurable rectangle in Ω is a set A =A1 × A2 × · · · × An, where Ai ∈ Ai for i = 1, 2, · · · , n. The smallest σ-algebra containing all measurable rectangles of Ω is called the product σ-algebra, denoted by A = A1 ×A2 × · · · ×An.

Note that the product σ-algebra A is the smallest σ-algebra containingmeasurable rectangles, rather than the product of A1,A2, · · · ,An.

Product σ-algebra may be easily extended to countably infinite case bydefining a measurable rectangle as a set of the form

A = A1 ×A2 × · · ·

where Ai ∈ Ai for all i and Ai = Ωi for all but finitely many i. The smallestσ-algebra containing all measurable rectangles of Ω = Ω1×Ω2× · · · is calledthe product σ-algebra, denoted by

A = A1 ×A2 × · · ·

Appendix B

Classical Measures

This appendix introduces the concepts of classical measure, measure space,Lebesgue measure, and product measure.

Definition B.1 Let Ω be a nonempty set, and A a σ-algebra over Ω. Aclassical measure π is a set function on A satisfyingAxiom 1. (Nonnegativity) πA ≥ 0 for any A ∈ A;Axiom 2. (Countable Additivity) For every countable sequence of mutuallydisjoint measurable sets Ai, we have

π

∞⋃i=1

Ai

=

∞∑i=1

πAi. (B.1)

Example B.1: Length, area and volume are instances of measure concept.

Definition B.2 Let Ω be a nonempty set, A a σ-algebra over Ω, and π ameasure on A. Then the triplet (Ω,A, π) is called a measure space.

It has been proved that there is a unique measure π on the Borel algebraof < such that π(a, b] = b− a for any interval (a, b] of <.

Definition B.3 The measure π on the Borel algebra of < such that

π(a, b] = b− a, ∀(a, b]

is called the Lebesgue measure.

Theorem B.1 (Product Measure Theorem) Let (Ωi,Ai, πi), i = 1, 2, · · · , nbe measure spaces such that πiΩi <∞ for i = 1, 2, · · · , n. Write

Ω = Ω1 × Ω2 × · · · × Ωn, A = A1 ×A2 × · · · ×An.

Appendix B - Classical Measures 217

Then there is a unique measure π on A such that

πA1 ×A2 × · · · ×An = π1A1 × π2A2 × · · · × πnAn (B.2)

for every measurable rectangle A1 × A2 × · · · × An. The measure π is calledthe product of π1, π2, · · · , πn, denoted by π = π1 × π2 × · · · × πn. The triplet(Ω,A, π) is called the product measure space.

Theorem B.2 (Infinite Product Measure Theorem) Assume that (Ωi,Ai, πi)are measure spaces such that πiΩi = 1 for i = 1, 2, · · · Write

Ω = Ω1 × Ω2 × · · · A = A1 ×A2 × · · ·

Then there is a unique measure π on A such that

πA1×· · ·×An×Ωn+1×Ωn+2×· · · = π1A1×π2A2×· · ·×πnAn (B.3)

for any measurable rectangle A1 × · · · × An × Ωn+1 × Ωn+2 × · · · and alln = 1, 2, · · · The measure π is called the infinite product, denoted by π =π1×π2×· · · The triplet (Ω,A, π) is called the infinite product measure space.

Appendix C

Measurable Functions

This appendix introduces the concepts of measurable function, simple func-tion, step function, absolutely continuous function, singular function, andCantor function.

Definition C.1 A function f from (Ω,A) to the set of real numbers is saidto be measurable if

f−1(B) = ω ∈ Ω|f(ω) ∈ B ∈ A (C.1)

for any Borel set B of real numbers. If Ω is a Borel set, then A is alwaysassumed to be the Borel algebra over Ω.

Theorem C.1 The function f is measurable from (Ω,A) to < if and only iff−1(I) ∈ A for any open interval I of <.

Definition C.2 A function f : < → < is said to be continuous if for anygiven x ∈ < and ε > 0, there exists a δ > 0 such that |f(y) − f(x)| < εwhenever |y − x| < δ.

Example C.1: Any continuous function f is measurable, because f−1(I) isan open set (not necessarily interval) of < for any open interval I ∈ <.

Example C.2: A monotone function f from < to < is measurable becausef−1(I) is an interval for any interval I.

Example C.3: A function is said to be simple if it takes a finite set ofvalues. A function is said to be step if it takes a countably infinite set ofvalues. Generally speaking, a step (or simple) function is not necessarilymeasurable except that it can be written as f(x) = ai if x ∈ Ai, where Ai

are measurable sets, i = 1, 2, · · · , respectively.

Appendix C - Measurable Functions 219

Example C.4: Let f be a measurable function from (Ω,A) to <. Then itspositive part and negative part

f+(ω) =

f(ω), if f(ω) ≥ 0

0, otherwise,f−(ω) =

−f(ω), if f(ω) ≤ 0

0, otherwise

are measurable functions, becauseω∣∣ f+(ω) > t

=ω∣∣ f(ω) > t

∪ω∣∣ f(ω) ≤ 0 if t < 0

,

ω∣∣ f−(ω) > t

=ω∣∣ f(ω) < −t

∪ω∣∣ f(ω) ≥ 0 if t < 0

.

Example C.5: Let f1 and f2 be measurable functions from (Ω,A) to <.Then f1 ∨ f2 and f1 ∧ f2 are measurable functions, because

ω∣∣ f1(ω) ∨ f2(ω) > t

=ω∣∣ f1(ω) > t

∪ω∣∣ f2(ω) > t

,

ω∣∣ f1(ω) ∧ f2(ω) > t

=ω∣∣ f1(ω) > t

∩ω∣∣ f2(ω) > t

.

Theorem C.2 Let fi be a sequence of measurable functions from (Ω,A)to <. Then the following functions are measurable:

sup1≤i<∞

fi(ω); inf1≤i<∞

fi(ω); lim supi→∞

fi(ω); lim infi→∞

fi(ω). (C.2)

Especially, if limi→∞ fi(ω) exists, then it is also a measurable function.

Theorem C.3 Let f be a nonnegative measurable function from (Ω,A) to<. Then there exists an increasing sequence hi of nonnegative simple mea-surable functions such that

limi→∞

hi(ω) = f(ω), ∀ω ∈ Ω. (C.3)

Furthermore, the functions hi may be defined as follows,

hi(ω) =

k − 1

2i, if

k − 12i

≤ f(ω) <k

2i, k = 1, 2, · · · , i2i

i, if i ≤ f(ω)(C.4)

for i = 1, 2, · · ·

Definition C.3 A function f : < → < is said to be Lipschitz continuous ifthere is a positive number K such that

|f(y)− f(x)| ≤ K|y − x|, ∀x, y ∈ <. (C.5)

220 Appendix C - Measurable Functions

Definition C.4 A function f : < → < is said to be absolutely continuous iffor any given ε > 0, there exists a small number δ > 0 such that

m∑i=1

|f(yi)− f(xi)| < ε (C.6)

for every finite disjoint class (xi, yi), i = 1, 2, · · · ,m of bounded open in-tervals for which

m∑i=1

|yi − xi| < δ. (C.7)

Definition C.5 A continuous and increasing function f : < → < is said tobe singular if f is not a constant and its derivative f ′ = 0 almost everywhere.

Example C.6: Let C be the Cantor set. We define a function g on C asfollows,

g

( ∞∑i=1

ai

3i

)=

∞∑i=1

ai

2i+1(C.8)

where ai = 0 or 2 for i = 1, 2, · · · Then g(x) is an increasing function andg(C) = [0, 1]. The Cantor function is defined on [0, 1] as follows,

f(x) = supg(y)

∣∣ y ∈ C, y ≤ x. (C.9)

It is clear that the Cantor function f(x) is increasing such that

f(0) = 0, f(1) = 1, f(x) = g(x), ∀x ∈ C.

Moreover, f(x) is a continuous function and f ′(x) = 0 almost everywhere.Thus the Cantor function f is a singular function.

Appendix D

Lebesgue Integral

This appendix introduces Lebesgue integral, Lebesgue-Stieltjes integral, mono-tone convergence theorem, Lebesgue dominated convergence theorem, andFubini theorem.

Definition D.1 Let h(x) be a nonnegative simple measurable function de-fined by

h(x) =

c1, if x ∈ A1

c2, if x ∈ A2

· · ·cm, if x ∈ Am

where A1, A2, · · · , Am are Borel sets. Then the Lebesgue integral of h on aBorel set A is ∫

A

h(x)dx =m∑

i=1

ciπA ∩Ai. (D.1)

Definition D.2 Let f(x) be a nonnegative measurable function on the Borelset A, and hi(x) a sequence of nonnegative simple measurable functionssuch that hi(x) ↑ f(x) as i→∞. Then the Lebesgue integral of f on A is∫

A

f(x)dx = limi→∞

∫A

hi(x)dx. (D.2)

Definition D.3 Let f(x) be a measurable function on the Borel set A, anddefine

f+(x) =

f(x), if f(x) > 0

0, otherwise,f−(x) =

−f(x), if f(x) < 0

0, otherwise.

Then the Lebesgue integral of f on A is∫A

f(x)dx =∫

A

f+(x)dx−∫

A

f−(x)dx (D.3)

222 Appendix D - Lebesgue Integral

provided that at least one of∫

Af+(x)dx and

∫Af−(x)dx is finite.

Definition D.4 Let f(x) be a measurable function on the Borel set A. Ifboth of

∫Af+(x)dx and

∫Af−(x)dx are finite, then the function f is said to

be integrable on A.

Theorem D.1 (Monotone Convergence Theorem) Let fi be an increasingsequence of measurable functions on A. If there is an integrable function gsuch that fi(x) ≥ g(x) for all i, then we have∫

A

limi→∞

fi(x)dx = limi→∞

∫A

fi(x)dx. (D.4)

Example D.1: The condition fi ≥ g cannot be removed in the monotoneconvergence theorem. For example, let fi(x) = 0 if x ≤ i and −1 otherwise.Then fi(x) ↑ 0 everywhere on A. However,∫

<lim

i→∞fi(x)dx = 0 6= −∞ = lim

i→∞

∫<fi(x)dx.

Theorem D.2 (Lebesgue Dominated Convergence Theorem) Let fi be asequence of measurable functions on A whose limitation limi→∞ fi(x) existsa.s. If there is an integrable function g such that |fi(x)| ≤ g(x) for any i,then we have ∫

A

limi→∞

fi(x)dx = limi→∞

∫A

fi(x)dx. (D.5)

Example D.2: The condition |fi| ≤ g in the Lebesgue dominated conver-gence theorem cannot be removed. Let A = (0, 1), fi(x) = i if x ∈ (0, 1/i)and 0 otherwise. Then fi(x) → 0 everywhere on A. However,∫

A

limi→∞

fi(x)dx = 0 6= 1 = limi→∞

∫A

fi(x)dx.

Theorem D.3 (Fubini Theorem) Let f(x, y) be an integrable function on<2. Then we have(a) f(x, y) is an integrable function of x for almost all y;(b) f(x, y) is an integrable function of y for almost all x;

(c)∫<2f(x, y)dxdy =

∫<

(∫<f(x, y)dy

)dx =

∫<

(∫<f(x, y)dx

)dy.

Theorem D.4 Let Φ be an increasing and right-continuous function on <.Then there exists a unique measure π on the Borel algebra of < such that

π(a, b] = Φ(b)− Φ(a) (D.6)

for all a and b with a < b. Such a measure is called the Lebesgue-Stieltjesmeasure corresponding to Φ.

Appendix D - Lebesgue Integral 223

Definition D.5 Let Φ(x) be an increasing and right-continuous function on<, and let h(x) be a nonnegative simple measurable function, i.e.,

h(x) =

c1, if x ∈ A1

c2, if x ∈ A2

...cm, if x ∈ Am.

Then the Lebesgue-Stieltjes integral of h on the Borel set A is∫A

h(x)dΦ(x) =m∑

i=1

ciπA ∩Ai (D.7)

where π is the Lebesgue-Stieltjes measure corresponding to Φ.

Definition D.6 Let f(x) be a nonnegative measurable function on the Borelset A, and let hi(x) be a sequence of nonnegative simple measurable func-tions such that hi(x) ↑ f(x) as i → ∞. Then the Lebesgue-Stieltjes integralof f on A is ∫

A

f(x)dΦ(x) = limi→∞

∫A

hi(x)dΦ(x). (D.8)

Definition D.7 Let f(x) be a measurable function on the Borel set A, anddefine

f+(x) =

f(x), if f(x) > 0

0, otherwise,f−(x) =

−f(x), if f(x) < 0

0, otherwise.

Then the Lebesgue-Stieltjes integral of f on A is∫A

f(x)dΦ(x) =∫

A

f+(x)dΦ(x)−∫

A

f−(x)dΦ(x) (D.9)

provided that at least one of∫

Af+(x)dΦ(x) and

∫Af−(x)dΦ(x) is finite.

Appendix E

Euler-Lagrange Equation

Let

L(φ) =∫ b

a

F (x, φ(x), φ′(x))dx, (E.1)

where F is a known function with continuous first and second partial deriva-tives. If L has an extremum (maximum or minimum) at φ(x), then

∂F

∂φ(x)− d

dx

(∂F

∂φ′(x)

)= 0 (E.2)

which is called the Euler-Lagrange equation. If φ′(x) is not involved, thenthe Euler-Lagrange equation reduces to

∂F

∂φ(x)= 0. (E.3)

Note that the Euler-Lagrange equation is only a necessary condition for theexistence of an extremum. However, if the existence of an extremum is clearand there exists only one solution to the Euler-Lagrange equation, then thissolution must be the curve for which the extremum is achieved.

Appendix F

Maximum UncertaintyPrinciple

An event has no uncertainty if its measure is 1 (or 0) because we may believethat the event occurs (or not). An event is the most uncertain if its measureis 0.5 because the event and its complement may be regarded as “equallylikely”.

In practice, if there is no information about the measure of an event,we should assign 0.5 to it. Sometimes, only partial information is available.For this case, the value of measure may be specified in some range. Whatvalue does the measure take? For the safety purpose, we should assign it thevalue as close to 0.5 as possible. This is the maximum uncertainty principleproposed by Liu [132].

Maximum Uncertainty Principle: For any event, if there are multiplereasonable values that a measure may take, then the value as close to 0.5 aspossible is assigned to the event.

Perhaps the reader would like to ask what values are reasonable. Theanswer is problem-dependent. At least, the values should ensure that allaxioms about the measure are satisfied, and should be consistent with thegiven information.

Example F.1: Let Λ be an event. Based on some given information, themeasure value MΛ is on the interval [a, b]. By using the maximum uncer-tainty principle, we should assign

MΛ =

a, if 0.5 < a ≤ b

0.5, if a ≤ 0.5 ≤ b

b, if a ≤ b < 0.5.

Appendix G

Uncertainty Relations

Probability theory is a branch of mathematics based on the normality, non-negativity, and countable additivity axioms. In fact, those three axiomsmay be replaced with four axioms: normality, monotonicity, self-duality, andcountable additivity. Thus all of probability, credibility, chance, and uncer-tain measures meet the normality, monotonicity and self-duality axioms. Theessential difference among those measures is how to determine the measureof union. For any mutually disjoint events Ai with supi πAi < 0.5, if πsatisfies the countable additivity axiom, i.e.,

π

∞⋃i=1

Ai

=

∞∑i=1

πAi, (G.1)

then π is a probability measure; if π satisfies the maximality axiom, i.e.,

π

∞⋃i=1

Ai

= sup

1≤i<∞πAi, (G.2)

then π is a credibility measure; if π satisfies the countable subadditivityaxiom, i.e.,

π

∞⋃i=1

Ai

≤

∞∑i=1

πAi, (G.3)

then π is an uncertain measure.Since additivity and maximality are special cases of subadditivity, prob-

ability and credibility are special cases of chance measure, and three of themare in the category of uncertain measure. This fact also implies that randomvariable and fuzzy variable are special cases of hybrid variables, and three ofthem are instances of uncertain variables.

Appendix G - Uncertainty Relations 227

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CredibilityModel

HybridModel

............................

.......................................

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...................................

ProbabilityModel

Uncertainty Model

Figure G.1: Relations among Uncertainties

Some Questions

Is the real world as extreme as probability model? Is the real world as extremeas credibility model? Is the real world as extreme as hybrid model? In manycases, the answer is negative. Perhaps uncertainty model is more realistic.

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List of Frequently Used Symbols

ξ, η, τ random, fuzzy, hybrid, or uncertain variablesξ, η, τ random, fuzzy, hybrid, or uncertain vectorsµ, ν membership functionsφ, ψ probability, or credibility density functionsΦ, Ψ probability, or credibility distributions

Pr probability measureCr credibility measureCh chance measureM uncertain measureE expected valueV varianceH entropy

(Γ,L,M) uncertainty space(Ω,A,Pr) probability space(Θ,P,Cr) credibility space

(Θ,P,Cr)× (Ω,A,Pr) chance space∅ empty set< set of real numbers<n set of n-dimensional real vectors∨ maximum operator∧ minimum operator

ai ↑ a a1 ≤ a2 ≤ · · · and ai → a

ai ↓ a a1 ≥ a2 ≥ · · · and ai → a

Ai ↑ A A1 ⊂ A2 ⊂ · · · and A = A1 ∪A2 ∪ · · ·Ai ↓ A A1 ⊃ A2 ⊃ · · · and A = A1 ∩A2 ∩ · · ·

Index

algebra, 213Borel algebra, 214Borel set, 214Brownian motion, 44canonical process, 207Cantor function, 220Cantor set, 214chance asymptotic theorem, 137chance density function, 145chance distribution, 143chance measure, 131chance semicontinuity law, 136chance space, 129chance subadditivity theorem, 134Chebyshev inequality, 33, 108, 158conditional chance, 165conditional credibility, 114conditional membership function, 116conditional probability, 40conditional uncertainty, 202convergence almost surely, 35, 110convergence in chance, 160convergence in credibility, 110convergence in distribution, 36, 111convergence in mean, 36, 110convergence in probability, 35countable additivity axiom, 1countable subadditivity axiom, 178credibility asymptotic theorem, 57credibility density function, 72credibility distribution, 69credibility extension theorem, 58credibility inversion theorem, 65credibility measure, 53credibility semicontinuity law, 57credibility space, 60credibility subadditivity theorem, 55critical value, 26, 96, 153, 194distance, 32, 106, 157, 196

entropy, 28, 100, 156, 195equipossible fuzzy variable, 68Euler-Lagrange equation, 224event, 1, 53, 130, 177expected value, 14, 80, 146, 187exponential distribution, 10exponential membership function, 95extension principle of Zadeh, 77Fubini theorem, 222fuzzy calculus, 124fuzzy differential equation, 128fuzzy integral equation, 128fuzzy process, 119fuzzy random variable, 129fuzzy sequence, 110fuzzy variable, 63fuzzy vector, 64hazard rate, 42, 118Holder’s inequality, 34, 108, 158, 198hybrid calculus, 171hybrid differential equation, 174hybrid integral equation, 174hybrid process, 169hybrid sequence, 160hybrid variable, 137hybrid vector, 142identification function, 184independence, 11, 74, 151, 193Ito formula, 48Ito integral, 47Ito process, 49Jensen’s inequality, 35, 109, 159, 199Lebesgue integral, 221Lebesgue measure, 216Lebesgue-Stieltjes integral, 223Lebesgue-Stieltjes measure, 222Lebesgue unit interval, 3Markov inequality, 33, 108, 158, 198maximality axiom, 54

246 Index

maximum entropy principle, 30, 103maximum uncertainty principle, 225measurable function, 218membership function, 65Minkowski inequality, 34, 109, 159moment, 25, 95, 151, 191monotone convergence theorem, 222monotone class theorem, 214monotonicity axiom, 53, 178nonnegativity axiom, 1normal distribution, 10normal membership function, 93normality axiom, 1, 53, 178optimistic value, see critical valuepessimistic value, see critical valuepower set, 213probability continuity theorem, 2probability density function, 9probability distribution, 7probability measure, 3probability space, 3product credibility axiom, 61product credibility space, 62product credibility theorem, 61product probability space, 4product probability theorem, 4random fuzzy variable, 129random sequence, 35random variable, 4random vector, 5renewal process, 43, 119, 169, 206self-duality axiom, 53, 178set function, 1σ-algebra, 213simple function, 218singular function, 220step function, 218stochastic differential equation, 50stochastic integral equation, 50stochastic process, 43stock model, 124, 171, 208trapezoidal fuzzy variable, 68triangular fuzzy variable, 68uncertain calculus, 209uncertain differential equation, 211uncertain integral equation, 211uncertain measure, 178uncertain process, 205

uncertain sequence, 200uncertain variable, 181uncertain vector, 182uncertainty asymptotic theorem, 180uncertainty density function, 185uncertainty distribution, 185uncertainty space, 181uniform distribution, 10variance, 23, 92, 149, 190

Wiener process, 44

uncertainty theory

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