Page 166 - DMTH404_STATISTICS

Statistics Richa Nandra, Lovely Professional University

Notes Unit 12: The Moment Generating Functions

CONTENTS

Objectives
Introduction
12.1 Moment Generating Function of a Random Variable
12.2 Deriving Moments with the mgf

12.3 Characterization of a Distriution via the mgf
12.4 Moment Generating Function - More details
12.4.1 Moment Generating Function of a Linear Transformation
12.4.2 Moment Generating Function of a Sum of Mutually Independent Random
Variable
12.5 Summary
12.6 Keywords

12.7 Self Assessment
12.8 Review Questions
12.9 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss moment generating function of a random variable
 Describe deriving moments with mgf

Introduction

In the previous unit, we learned that the expected value of the sample mean X is the population
2
mean m. We also learned that the variance of the sample mean X is  /n, that is, the population
variance divided by the sample size n. We have not yet determined the probability distribution
of the sample mean when, say, the random sample comes from a normal distribution with mean
2
m and variance  . We are going to tackle that in the next lesson! Before we do that, though, we
are going to want to put a few more tools into our tollbox. We already have learned a few
techniques for finding the probability distribution of a function of random variables, namely
the distribution function technique and the change-of-variable technique. In this unit, we’ll
learn yet another technique called the moment-generating function technique. We’ll use the
technique in this lesson to learn, among other things, the distribution of sums of chi-square
random variables, then, in the next lesson, we’ll use the technique to find (finally) the probability
distribution of the sample mean when the random sample comes from a normal distribution
2
with mean m and variance  .

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