Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.5 PROPERTIES OF DISTRIBUTIONS

|x−μ|≥c. From (30.48), we find that

σ^2 ≥

∫

|x−μ|≥c

(x−μ)^2 f(x)dx≥c^2

∫

|x−μ|≥c

f(x)dx. (30.49)

The first inequality holds because both (x−μ)^2 andf(x) are non-negative for

allx, and the second inequality holds because (x−μ)^2 ≥c^2 over the range of

integration. However, the RHS of (30.49) is simply equal toc^2 Pr(|X−μ|≥c),

and thus we obtain the required inequality

Pr(|X−μ|≥c)≤

σ^2

c^2

.

A similar derivation may be carried through for the case of a discrete random

variable. Thus, foranydistributionf(x) that possesses a variance we have, for

example,

Pr(|X−μ|≥ 2 σ)≤

1

4

and Pr(|X−μ|≥ 3 σ)≤

1

9

.

30.5.4 Moments

The mean (or expectation) ofXis sometimes called thefirst momentofX,since

it is defined as the sum or integral of the probability density function multiplied

by the first power ofx. By a simple extension thekth moment of a distribution

is defined by

μk≡E[Xk]=

{∑

jx

k

jf(xj) for a discrete distribution,

∫

xkf(x)dx for a continuous distribution. (30.50)

For notational convenience, we have introduced the symbolμkto denoteE[Xk],

thekth moment of the distribution. Clearly, the mean of the distribution is then

denoted byμ 1 , often abbreviated simply toμ, as in the previous subsection, as

this rarely causes confusion.

A useful result that relates the second moment, the mean and the variance of

a distribution is proved using the properties of the expectation operator:

V[X]=E

[

(X−μ)^2

]

=E

[

X^2 − 2 μX+μ^2

]

=E

[

X^2

]

− 2 μE[X]+μ^2

=E

[

X^2

]

− 2 μ^2 +μ^2

=E

[

X^2

]

−μ^2. (30.51)

In alternative notations, this result can be written

〈(x−μ)^2 〉=〈x^2 〉−〈x〉^2 or σ^2 =μ 2 −μ^21.