Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

96 Multivariate Distributions

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x

y

( 0 , y )( y, y )

Figure 2.1.3: Region of integration for Example 2.1.8. The arrow depicts the

integration with respect tox 1 at a fixed but arbitraryx 2.

For the second way, we make use of expression (2.1.10) and findE(Y) directly by

E(Y)=E

(

X 1

X 2

)

=

∫ 1

0

{∫x 2

0

(

x 1

x 2

)

8 x 1 x 2 dx 1

}

dx 2

=

∫ 1

0

8

3

x^32 dx 2 =

2

3

.

We next define the moment generating function of a random vector.

Definition 2.1.2(Moment Generating Function of a Random Vector). LetX=
(X 1 ,X 2 )′ be a random vector. IfE(et^1 X^1 +t^2 X^2 )exists for|t 1 |<h 1 and|t 2 |<
h 2 ,whereh 1 andh 2 are positive, it is denoted byMX 1 ,X 2 (t 1 ,t 2 )and is called the
moment generating function(mgf) ofX.

As in the one-variable case, if it exists, the mgf of a random vector uniquely
determines the distribution of the random vector.
Lett=(t 1 ,t 2 )′. ThenwecanwritethemgfofXas

MX 1 ,X 2 (t)=E

[

et

′X]

, (2.1.13)

so it is quite similar to the mgf of a random variable. Also, the mgfs ofX 1 andX 2
are immediately seen to beMX 1 ,X 2 (t 1 ,0) andMX 1 ,X 2 (0,t 2 ), respectively. If there
is no confusion, we often drop the subscripts onM.