lecture23.pdf
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Lecture 23
Zhihua (Sophia) Su
University of Florida
Mar 13, 2015
STA 4321/5325 Introduction to Probability 1
Agenda
Moment Generating Function (MGF)PropertiesMixed Random Variable
Reading assignment: Chapter 3.9, Chapter4: 4.9, 4.11
STA 4321/5325 Introduction to Probability 2
Moment Generating Function (MGF)
In the last lecture, we defined the notion of a momentgenerating functions. Today, we will derive the momentgenerating functions for some standard random variables, andlearn a very useful property of moment generating functions.
STA 4321/5325 Introduction to Probability 3
Moment Generating Function (MGF)
If X is Binomial (n, p), derive its MGF.
STA 4321/5325 Introduction to Probability 4
Moment Generating Function (MGF)
If Z is Normal (0, 1), derive its MGF.
STA 4321/5325 Introduction to Probability 5
Moment Generating Function (MGF)
If X is Gamma (α, β), derive its MGF.
STA 4321/5325 Introduction to Probability 6
Properties
Property of MGF
If X and Y are two random variables such that
MX(t) =MY (t) for every t ∈ R,
(assuming that MX(t) <∞ for an interval around 0), then Xand Y have the same distribution.
STA 4321/5325 Introduction to Probability 7
Properties
Application: Let Z be standard normal. We are interested infinding the distribution of Z2.
STA 4321/5325 Introduction to Probability 8
Mixed Random Variable
Until now, we have studied two kinds of random variables,discrete and continuous. Here is a characterization of discreteand continuous random variables in terms of their distributionfunctions.
ResultA random variable X is discrete if and only if its distributionfunction is a piecewise constant function with positive jumps atpoints in X =Range(X).
ResultA random variable X is continuous if and only if its distributionfunction is a continuous function (piecewise differentiable).
STA 4321/5325 Introduction to Probability 9
Mixed Random Variable
However, there are many random variables, whose distributionfunctions do not behave in either of the two ways written above.Such random variables are called mixed random variables.
Here is a simple example of a mixed random variable.Let X denote the life length (in hundreds of hour) of a certaintype of electronic component. The components frequently failimmediately upon insertion into the system. The probability ofimmediate failure is 1
4 . However, if a component does not failimmediately, its life-length distribution has the exponentialdensity
fX(x) =
{e−x if x > 0,
0 if x ≤ 0.
STA 4321/5325 Introduction to Probability 10