Lecture 23

Zhihua (Sophia) Su

University of Florida

Mar 13, 2015

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Agenda

Moment Generating Function (MGF)PropertiesMixed Random Variable

Reading assignment: Chapter 3.9, Chapter4: 4.9, 4.11

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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.

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Moment Generating Function (MGF)

If X is Binomial (n, p), derive its MGF.

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Moment Generating Function (MGF)

If Z is Normal (0, 1), derive its MGF.

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Moment Generating Function (MGF)

If X is Gamma (α, β), derive its MGF.

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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.

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Properties

Application: Let Z be standard normal. We are interested infinding the distribution of Z2.

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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).

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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.

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