Page 44 - Elements of Distribution Theory

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30 Properties of Probability Distributions

Hence,

E(Y) = yp(x) = g(x)p(x).
y x:g(x)=y y x:g(x)=y
Since every x value in the range of X leads to some value y in the range of Y,it follows that

E(Y) = g(x)p(x).
x
In the general case, the result is simply the change-of-variable formula for integration, as
discussed in Appendix 1; see, for example, Billingsley (1995, Theorem 16.13) for a proof.
The result (1.3) is usually expressed without reference to the random variable Y:

E[g(X)] = g(x) dF(x)
X
provided that the integral exists. Note that it causes no problem if g(x)is undefined for x ∈ A
for some set A such that Pr(X ∈ A) = 0; this set can simply be omitted when computing
the expected value.

Example 1.31 (Standard exponential distribution). Let X denote a random variable with
a standard exponential distribution; see Example 1.16. Then X has density function
p(x) = exp(−x), 0 < x < ∞.

r
r
Consider the expected value of X where r > 0isa constant. If E(X )exists, it is given by
∞

r
x exp(−x) dx,
0
which is simply the well-known gamma function evaluated at r + 1, (r + 1); the gamma
function is discussed in detail in Section 10.2.
r
Expected values of the form E(X ) for r = 1, 2,... are called the moments of the dis-
tribution or the moments of X. Thus, the moments of the standard exponential distribution
are r!, r = 1, 2,.... Moments will be discussed in detail in Chapter 4.

Example 1.32 (Uniform distribution on the unit cube). Let X denote a three-dimensional
3
random vector with the uniform distribution on (0, 1) ; see Example 1.30. Let Y = X 1 X 2 X 3 ,
where X = (X 1 , X 2 , X 3 ). Then
1
1 1 1
E(Y) = x 1 x 2 x 3 dx 1 dx 2 dx 3 = .
0 0 0 8
Given the correspondence between E[g(X)] and integrals of the form

g(x) dF(x),
X
many important properties of expectation may be derived directly from the corresponding
properties of integrals, given in Appendix 1. Theorem 1.10 contains a number of these; the
proof follows immediately from the results in Appendix 1 and, hence, it is omitted.