Fundamentals of Probability and Statistics for Engineers

4 EXPECTATIONS AND MOMENTS

While a probability distribution [FX (x), pX (x), or f (^) X(x)] co ntains a complete
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 in clude moments of various orders associated with X. Let us fir st
provide a general definition (Definition 4.1).
D ef inition 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
is defined by
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,
The symbol is regarded here and in the sequel as the expectation operator.
If random variable X is continuous, the expectation is defined by
if the improper integral is absolutely convergent, that is,
Fundamentals of Probability and Statistics for Engineers T.T. Soong  2004 John Wiley & Sons, Ltd
ISBN s: 0-470-86813-9 (H B) 0-470-86814-7 (PB)
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