Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

112 Multivariate Distributions

and

f 2 (x 2 )=

{∫x 2

0 2 dx^1 =2x^20 <x^2 <^1

0elsewhere.

The conditional pdf ofX 1 ,givenX 2 =x 2 , 0 <x 2 <1, is

f 1 | 2 (x 1 |x 2 )=

{ 2

2 x 2 =

1

x 2 0 <x^1 <x^2 <^1

0elsewhere.

Here the conditional mean and the conditional variance ofX 1 ,givenX 2 =x 2 ,are
respectively,

E(X 1 |x 2 )=

∫∞

−∞

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

=

∫x 2

0

x 1

(

1

x 2

)

dx 1

=

x 2

2

, 0 <x 2 < 1 ,

and

Var(X 1 |x 2 )=

∫x 2

0

(

x 1 −

x 2

2

) 2 ( 1

x 2

)

dx 1

=

x^22

12

, 0 <x 2 < 1.

Finally, we compare the values of

P(0<X 1 <^12 |X 2 =^34 )andP(0<X 1 <^12 ).

We have

P(0<X 1 <^12 |X 2 =^34 )=

∫ 1 / 2

0

f 1 | 2 (x 1 |^34 )dx 1 =

∫ 1 / 2

0

(^43 )dx 1 =^23 ,

but

P(0<X 1 <^12 )=

∫ 1 / 2

0 f^1 (x^1 )dx^1 =

∫ 1 / 2

0 2(1−x^1 )dx^1 =

3

4.

SinceE(X 2 |x 1 ) is a function ofx 1 ,thenE(X 2 |X 1 ) is a random variable with its

own distribution, mean, and variance. Let us consider the following illustration of

this.

Example 2.3.2.LetX 1 andX 2 have the joint pdf

f(x 1 ,x 2 )=

{

6 x 2 0 <x 2 <x 1 < 1

0elsewhere.

Then the marginal pdf ofX 1 is

f 1 (x 1 )=

∫x 1

0

6 x 2 dx 2 =3x^21 , 0 <x 1 < 1 ,