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.