probability theory lecture notes 01

8/8/2019 Probability Theory Lecture notes 01

http://slidepdf.com/reader/full/probability-theory-lecture-notes-01 1/3

Figure 1: Three sets can generate 7 atoms.

Outline

Contents

1 Set and Functions 1

1 Set and Functions

Set Operations

• The whole set Ω, the empty set φ; an element x in a set A: x ∈ A.

• A ⊂ B, A ⊆ B; A ⊃ B, A ⊇ B.

• Set operations: A ∩ B, A ∪B, Ac (w.r.t. Ω).

• De Morgan’s laws: (A ∪ B)c

= Ac ∩Bc, (A ∩B)c

= Ac ∪Bc.

• For more than two sets:n An

c

=n Ac

n,

n An

c

=n Ac

n.

Disjoint “atoms”: the classical three-circle-diagram.



Functions

• Please review the basic definitions of a function. Pay attention to a function’sdomain and its image.

• I assume you know the definition and basic properties of the following elemen-

tary functions:

1. Power functions, xa, a ∈ R. When a is a non-integer rational number,

without confusion we take the principle branch of nth root operation; when

a is irrational, its definition is given by the principle branch of a log func-

tion.

2. Exponential and logarithmic functions, ex and log(x).

3. Trigonometric functions and their inverse functions. Their domain, image,

etc.

4. The combination of the above by +,×,÷, and functional composition.

Limit of a sequence of real numbers (I)

• For simplicity, we are going to use “increasing” to mean “non-decreasing”. “de-

creasing” to mean “non-increasing”.

• A sequence of real numbers (ai) = (a1, a2, . . .) converges to a∗ if for any given

“precision criterion” > 0, there exists an integer N such that the “error”, de-

fined as dist(ai − a∗) = |ai − a∗|, is smaller than for all i ≥ N .

• An increasing, bounded sequence of real numbers a1 ≤ a2 ≤ . . . always con-

verges to a limit. (if you consider “∞” as a valid limiting point, the boundedness

part can be omitted.)

• Similarly, a decreasing sequence of real numbers always converges to a limit (if

you don’t like “−∞”, you can add the bounded from below condition).

Limit of a sequence of real numbers (II)

• In general a sequence of real numbers (a1, a2, . . .) always has a subsequence

which approaches the upper limit (including “∞” as a possible limit) of this

sequence.

• Similarly, it contains a subsequence which approaches its lower limit . Once the

upper limit equals the lower limit, we say this sequence converges to this limit.

• Two companion subsequences, denoted as (b1, b2, . . .) and (c1, c2, . . .), can be

quite useful:

bi = supi≥n

ai, ci = inf i≥n

ai.

• (bi) is decreasing and (ci) is increasing and they converge to lim supi→∞ ai and

liminf i→∞ ai, respectively.

2



Limit of a sequence of real numbers (IV)

• For a sequence (an), If its upper limit equals its lower limit (lim supnan =liminf nan = a∗), then (an) converges to a.

• The distance function which quantifies error is important. For real numbers,

there is essentially one way to measure the error term: |ai− a∗|. This is because

a distance function needs to satisfy several axioms (use wikipedia).

• For a sequence of n-dimensional points(vectors), the natural way to measure

the error term is the Euclidean distance. But other distance functions do exist,

such as the Manhattan distance (google it). Fortunately, a sequence of vectors is

convergent in one distance implies it is convergent in all other distances.

• Unfortunately, you can define quite a few non-compatible distances of random

numbers. So there are many different convergences of random numbers.

Limit of a function

• We can now define the limit of a function f (x) as x approaches x0. x0 could be

±∞, for the sake of simplicity we assume x0 is finite for the following definition.

• limx→x0f (x) = y∗ if and only if

∀ > 0, ∃δ > 0 such that dist(f (x)− y∗) < for all x ∈ Ball(x0, δ).

• The logic negation of a sequence/function converges to a value is that this se-

quence/function “breaks the precision rule” infinitely often. More precisely, a

sequence is not convergent if for a given > 0, we have

dist(ai, a∗) > i.o.

where i.o. stands for infinitely often.

3

probability theory lecture notes 01

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