Page 92 - Fundamentals of Probability and Statistics for Engineers

4

Expectations and Moments

While a probability distribution [F X (x), p (x), or f (x)] contains a complete
X
X
description of a random variable X, it is often of interest to seek a set of simple
numbers that gives the random variable some of its dominant features. These
numbers include moments of various orders associated with X. Let us first
provide a general definition (Definition 4.1).

Definition 4.1. Let g(X ) be a real-valued function of a random variable X.

The mathematical expectation, or simply expectation, of g(X ), denoted by
Efg X)g, is defined by
X
Efg X†g ˆ g x i †p x i †; 4:1†
X
i
if X is discrete, where x 1 , x 2 ,.. . are possible values assumed by X.
When the range of i extends from 1 to infinity, the sum in Equation (4.1)
exists if it converges absolutely; that is,
1
X
jg x i †jp x i † < 1:
X
iˆ1

The symbol Efg is regarded here and in the sequel as the expectation operator.
If random variable X is continuous, the expectation Efg X)g is defined by

Z 1
Efg X†g ˆ g x†f x†dx; 4:2†
X
1
if the improper integral is absolutely convergent, that is,
Z 1
jg x†jf x†dx < 1:
X
1
Fundamentals of Probability and Statistics for Engineers T.T. Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)

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