Page 108 - Elements of Distribution Theory

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4

Moments and Cumulants

4.1 Introduction
Let X denote a real-valued random variable. As noted in Chapter 1, expected values of
r
the form E(X ), for r = 1, 2,... , are called the moments of X. The moments of a random
variable give one way to describe its properties and, in many cases, the sequence of moments
uniquely determines the distribution of the random variable. The most commonly used
moment is the first moment, E(X), called the mean of the distribution or the mean of X.In
elementary statistics courses, the mean of a distribution is often described as a “measure of
central tendency.”
Let µ = E(X). The central moments of X are the moments of X − µ. The most com-
2
monly used central moment is the second central moment, E[(X − µ) ], called the variance.
The variance is a measure of the dispersion of the distribution around its mean.
In this chapter, we consider properties of moments, along with associated quantities such
as moment-generating functions and cumulants, certain functions of the moments that have
many useful properties.

4.2 Moments and Central Moments
Let X denote a real-valued random variable. It is important to note that, for a given value
r
of r,E(X ) may be infinite, or may not exist. As with any function of X,if
r r
E(|X |) ≡ E(|X| ) < ∞,
r
r
then E(X )exists and is finite. If, for some r,E(|X| ) < ∞, then,
j
E(|X| ) < ∞, j = 1, 2,...,r.
m
This follows from Jensen’s inequality, using the fact that the function t , t > 0, is convex
for m ≥ 1; then,
r
r
r
j
j
E(|X| ) ≤ E[(|X| ) ] = E(|X| ) < ∞.
j
j
j
so that E(|X| ) < ∞.
Example 4.1 (Standard exponential distribution). In Example 1.31 it was shown that if
X has a standard exponential distribution, then
r
E(X ) = (r + 1), r > 0.
Hence, the moments of X are r!, r = 1, 2,....
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