lecture22 (1).pdf

Lecture 22

Zhihua (Sophia) Su

University of Florida

Mar 11, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Beta Random VariablesMoment Generating Function

Reading assignment: Chapter 4: 4.7, 4.8,4.9


Beta Random Variables

Every continuous distribution that we have encountered, exceptthe uniform, takes values over an infinite interval ((0,∞) or R).The beta distribution is an alternative model for randomvariable which are constrained to lie in the interval (0, 1).

A random variable X is said to have a Beta(α, β) distribution(where α and β are fixed constants) if

X = Range(X) = (0, 1),

fX(x) =

{Γ(α+β)

Γ(α)Γ(β)xα−1(1− x)β−1 if 0 ≤ x ≤ 1,

0 if x 6∈ [0, 1].



The first thing that we need to check isˆ ∞−∞

fX(x)dx = 1.

To prove this, we use the following identity.

ResultIf α, β are positive constants, then

ˆ 1

0xα−1(1− x)β−1dx =

Γ(α)Γ(β)

Γ(α+ β).



The distribution function FX is not available in closed form, butcan be evaluated using various software packages. The Betadensity can take a variety of shapes which points to theflexibility of this collection of random variables.

ResultThe Uniform [0, 1] random variable is a special case of the Betarandom variable with α = 1, β = 1.



Find the expected value E(X) and variance V (X) for a Betarandom variable X.



Example: A gasoline wholesale distributor uses bulk storagetanks to hold a fixed supply. The tanks are filled every Monday.Of interest to the wholesaler is the proportion of the supply soldduring the week. For many weeks, this proportion has beenobserved to match fairly well a Beta distribution with α = 4 andβ = 2.

(a) Find the expected value of this proportion.

(b) Find the probability that the wholesaler will sell at least90% of the stock in a given week.


Moment Generating Function

For a random variable X (discrete or continuous), the momentgenerating function is a special function associated to it. Themoment generating function is a powerful theoretical tool. Thename “moment generating function” comes from the fact that itcan be used to obtain the “moments” of the random variable.We clarify this shortly.



DefinitionIf X is a discrete or continuous random variable, then themoment generating function of X is denoted by MX :R→ (0,∞), and is defined by

MX(t) = E(etx) =

{∑x∈X etxP (X = x) if x is discrete,´∞−∞ e

txfX(x)dx if x is continuous.

Important note: The moment generating function is definedfor only those t, where the sums or integrals considered in thedefinition exist.


Moment Generating FunctionFor a random variable X, the values

E(X),E(X2),E(X3), · · ·are known as the moments of the random variable. Theyprovide useful information about the random variable.

Result

d

dtMX(t)

∣∣∣t=0

= E(X),

d2

dt2MX(t)

∣∣∣t=0

= E(X2).

In fact, for any positive integer k,

dk

dtkMX(t)

∣∣∣t=0

= E(Xk).



Hence the moment generating function can be used to computethe moments of the random variable (provided that the momentgenerating function is well defined in an interval around zero).


lecture22 (1).pdf

Documents