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Statistics Sachin Kaushal, Lovely Professional University

Notes Unit 13: Moment Generating Function - Continue

CONTENTS

Objectives
Introduction
13.1 Joint Moment Generating Function
13.2 Properties of Moment Generating Function

13.3 Summary
13.4 Keywords
13.5 Self Assessment
13.6 Review Questions

13.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the joint moment generating function
 Describe properties of moment generating function

Introduction

In probability theory and statistics, the moment-generating function of any random variable is
an alternative definition of its probability distribution. Thus, it provides the basis of an alternative
route to analytical results compared with working directly with probability density functions
or cumulative distribution functions. There are particularly simple results for the moment-
generating functions of distributions defined by the weighted sums of random variables.
In addition to univariate distributions, moment-generating functions can be defined for vector-
or matrix-valued random variables, and can even be extended to more general cases.
The moment-generating function does not always exist even for real-valued arguments, unlike
the characteristic function. There are relations between the behavior of the moment-generating
function of a distribution and properties of the distribution, such as the existence of moments.

13.1 Joint Moment Generating Function

we start this lecture by defining the moment generating function of a random vector.

Definition Let X be a K × 1 random vector. If the expected value
E[exp(t X) = E[exp(t X + t X + ... t X )]
T
1 1 2 2 K K
exists and is finite for all k × 1 real vectors t belonging to a closed rectangle H:
H = [–h , h ] × [–h , h ] × ... × [–h , h ]   K
1 1 2 2 K K

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