Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.1. Distributions of Two Random Variables 97

Example 2.1.10.Let the continuous-type random variablesXandY have the

joint pdf

f(x, y)=

{

e−y 0 <x<y<∞

0elsewhere.

The reader should sketch the space of (X, Y). The mgf of this joint distribution is

M(t 1 ,t 2 )=

∫∞

0

[∫∞

x

exp(t 1 x+t 2 y−y)dy

]

dx

=

1

(1−t 1 −t 2 )(1−t 2 )

,

provided thatt 1 +t 2 <1andt 2 <1. Furthermore, the moment-generating func-
tions of the marginal distributions ofXandYare, respectively,

M(t 1 ,0) =

1

1 −t 1

,t 1 < 1

M(0,t 2 )=

1

(1−t 2 )^2

,t 2 < 1.

These moment-generating functions are, of course, respectively, those of the
marginal probability density functions,

f 1 (x)=

∫∞

x

e−ydy=e−x, 0 <x<∞,

zero elsewhere, and

f 2 (y)=e−y

∫y

0

dx=ye−y, 0 <y<∞,

zero elsewhere.
We also need to define the expected value of the random vector itself, but this
is not a new concept because it is defined in terms of componentwise expectation:

Definition 2.1.3(Expected Value of a Random Vector).LetX=(X 1 ,X 2 )′be a
random vector. Then theexpected valueofXexists if the expectations ofX 1 and
X 2 exist. If it exists, then theexpected valueis given by

E[X]=

[

E(X 1 )

E(X 2 )

]

. (2.1.14)

EXERCISES
2.1.1.Letf(x 1 ,x 2 )=4x 1 x 2 , 0 <x 1 < 1 , 0 <x 2 <1, zero elsewhere, be the pdf
ofX 1 andX 2 .FindP(0<X 1 <^12 ,^14 <X 2 <1),P(X 1 =X 2 ),P(X 1 <X 2 ), and
P(X 1 ≤X 2 ).
Hint:Recall thatP(X 1 =X 2 ) would be the volume under the surfacef(x 1 ,x 2 )=
4 x 1 x 2 and above the line segment 0<x 1 =x 2 <1inthex 1 x 2 -plane.