Page 137 - Probability and Statistical Inference

114 3. Multivariate Random Variables

Similarly, the Theorem 3.3.1 (ii) will imply that

which reduces to 330. Refer to the Sections 2.3.2 and 2.3.4 as needed. In a
situation like this, again the marginal distribution of X is referred to as a
2
compound distribution. Note that we have been able to derive the expressions
of the mean and variance of X without first identifying the marginal distribu-
2
tion of X . !
2
Example 3.3.8 Suppose that conditionally given X = x , the random vari-
1
1
able X is distributed as Binomial(n, x ) for any fixed x ∈ (0, 1). Suppose also
1
1
2
that marginally, X is distributed as Beta(α = 4, β = 6). How should we pro-
1
ceed to find E[X ] and V[X ]? The Theorem 3.3.1 (i) will immediately imply
2
2
that
Similarly, the Theorem 3.3.1 (ii) will imply that

Refer to the Sections 2.3.1 and equation (1.7.35) as needed. One can verify
that Also, we have .
Hence,
we have

In a situation like this, again the marginal distribution of X is referred to as a
2
compound distribution. Note that we have been able to derive the expressions
of the mean and variance of X without first identifying the marginal distribu-
2
tion of X . !
2
Example 3.3.9 (Example 3.3.2 Continued) Let us now apply the Theo-
rem 3.3.1 to reevaluate the expressions for E[X ] and V[X ]. Combining
2
2
(3.3.15) with the Theorem 3.3.1 (i) and using direct integration with respect