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3. Multivariate Random Variables 113

But, observe that the third term in the last step (3.3.29) can be simplified as
follows:

which is zero. Now, we combine (3.3.29)-(3.3.30) and obtain

At this point, (3.3.31) combined with (3.3.29) then leads to the desired result
stated in part (ii). ¢
The next three Examples 3.3.6-3.3.8, while applying
the Theorem 3.3.1, introduce what are otherwise
known in statistics as the compound distributions.

Example 3.3.6 Suppose that conditionally given X = x , the random vari-
1 1
able X is distributed as N(β + β x , ) for any fixed x ∈ ℜ. Here, β , β are
2 0 1 1 1 0 1
two fixed real numbers. Suppose also that marginally, X is distributed as N(3,
1
10). How should we proceed to find E[X ] and V[X ]? The Theorem 3.3.1 (i)
2
2
will immediately imply that

which reduces to β + 3β . Similarly, the Theorem 3.3.1 (ii) will imply that
0 1

which reduces to Refer to the Section 2.3.3 as needed. In a situa-
tion like this, the marginal distribution of X is referred to as a compound
2
distribution. Note that we have been able to derive the expressions of the
mean and variance of X without first identifying the marginal distribution of
2
X . !
2
Example 3.3.7 Suppose that conditionally given X = x , the random vari-
1
1
+
able X is distributed as Poisson (x ) for any fixed x ∈ ℜ . Suppose also that
1
1
2
marginally, X is distributed as Gamma (α = 3, β = 10). How should we
1
proceed to find E[X ] and V[X ]? The Theorem 3.3.1 (i) will immediately
2
2
imply that