Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.3. Conditional Distributions and Expectations 111

We often abbreviate these conditional pdfs byf 1 | 2 (x 1 |x 2 )andf 2 | 1 (x 2 |x 1 ), respec-
tively. Similarly,f 1 (x 1 )andf 2 (x 2 ) are used to denote the respective marginal pdfs.
Since each off 2 | 1 (x 2 |x 1 )andf 1 | 2 (x 1 |x 2 ) is a pdf of one random variable, each
has all the properties of such a pdf. Thus we can compute probabilities and math-
ematical expectations. If the random variables are of the continuous type, the
probability

P(a<X 2 <b|X 1 =x 1 )=

∫b

a

f 2 | 1 (x 2 |x 1 )dx 2

is called “the conditional probability thata<X 2 <b,giventhatX 1 =x 1 .” If there
is no ambiguity, this may be written in the formP(a<X 2 <b|x 1 ). Similarly, the
conditional probability thatc<X 1 <d,givenX 2 =x 2 ,is

P(c<X 1 <d|X 2 =x 2 )=

∫d

c

f 1 | 2 (x 1 |x 2 )dx 1.

Ifu(X 2 ) is a function ofX 2 ,theconditional expectationofu(X 2 ), given that
X 1 =x 1 , if it exists, is given by

E[u(X 2 )|x 1 ]=

∫∞

−∞

u(x 2 )f 2 | 1 (x 2 |x 1 )dx 2.

Note thatE[u(X 2 )|x 1 ] is a function ofx 1. If they do exist, thenE(X 2 |x 1 )isthe
mean andE{[X 2 −E(X 2 |x 1 )]^2 |x 1 }is theconditional varianceof the conditional
distribution ofX 2 ,givenX 1 =x 1 , which can be written more simply as Var(X 2 |x 1 ).
It is convenient to refer to these as the “conditional mean” and the “conditional
variance” ofX 2 ,givenX 1 =x 1 .Ofcourse,wehave

Var(X 2 |x 1 )=E(X 22 |x 1 )−[E(X 2 |x 1 )]^2

from an earlier result. In a like manner, the conditional expectation ofu(X 1 ), given
X 2 =x 2 , if it exists, is given by

E[u(X 1 )|x 2 ]=

∫∞

−∞

u(x 1 )f 1 | 2 (x 1 |x 2 )dx 1.

With random variables of the discrete type, these conditional probabilities and
conditional expectations are computed by using summation instead of integration.
An illustrative example follows.

Example 2.3.1.LetX 1 andX 2 have the joint pdf

f(x 1 ,x 2 )=

{

20 <x 1 <x 2 < 1

0elsewhere.

Then the marginal probability density functions are, respectively,

f 1 (x 1 )=

{ ∫ 1

x 12 dx^2 =2(1−x^1 )0<x^1 <^1

0elsewhere,