Page 96 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 79

Let us note that the mean life, EfTg, is given by
Z 1 1
EfTgˆ tf t†dt ˆ :
T
0
A point x i such that

p x i † > p x i‡1 † and p x i † > p x i 1 †; X discrete;
X
X
X
X
f x i † > f x i ‡ "† and f x i † > f x i "†; X continuous;
X
X
X
X
"
where is an arbitrarily small positive quantity, is called a mode of X. A mode is
thus a value of X corresponding to a peak in its mass function or density
function. The term unimodal distribution refers to a probability distribution
possessing a unique mode.
To give a comparison of these three measures of centrality of a random
variable, Figure 4.1 shows their relative positions in three different situations. It
is clear that the mean, the median, and the mode coincide when a unimodal
distribution is symmetric.
4.1.2 CENTRAL MOMENTS, VARIANCE, AND STANDARD
DEVIATION

Besides the mean, the next most important moment is the variance, which
measures the dispersion or spread of random variable X about its mean. Its
definition will follow a general definition of central moments (see Definition 4.2).

Definition 4.2. The central moments of random variable X are the moments of

X with respect to its mean. Hence, the nth central moment of X , n ,is defined as
n X n
n ˆ Ef X m† gˆ x i m† p x i †; X discrete; 4:6†
X
i
Z 1
n
n
n ˆ Ef X m† gˆ x m† f x†dx; X continuous: 4:7†
X
1
The variance of X is the second central moment, 2 , commonly denoted by 2 X

or simply 2 or var(X ). It is the most common measure of dispersion of
a distribution about its mean. Large values of 2 X imply a large spread in
the distribution of X about its mean. Conversely, small values imply a sharp
concentration of the mass of distribution in the neighborhood of the mean. This is
illustrated in Figure 4.2 in which two density functions are shown with the same
2
mean but different variances. When ˆ 0, the whole mass of the distribution is
X
concentrated at the mean. In this extreme case, X ˆ m X with probability 1.

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