Page 135 - Probability and Statistical Inference

112 3. Multivariate Random Variables

and hence using (3.3.24), one can write

One may, for example, evaluate µ too given that X = .5. We leave it out as
1
2/1
the Exercise 3.3.3. !
We defined the conditional mean and variance of a random variable X
given the other random variable X in (3.3.7). The following result gives the 1
2
tool for finding the unconditional mean and variance of X utilizing the expres-
1
sions of the conditional mean and variance.
Theorem 3.3.1 Suppose that X = (X , X ) has a bivariate pmf or pdf,
2
1
namely f(x , x ) for x ∈ χ , the support of X , i = 1, 2. Let E [.] and V [.]
1
1
i
2
1
i
i
respectively denote the expectation and variance with respect to the marginal
distribution of X . Then, we have
1
(i) E[X ] = E [E {X | X }];
2 1 2/1 2 1
(ii) V(X ) = V [E {X | X }] + E [V /V {X | X }].
2 1 2/1 2 1 1 1 2/1 2 1
Proof (i) Note that
Next, we can rewrite

which is the desired result.
(ii) In view of part (i), we can obviously write

where g(x ) = E{(X – χ ) | X = x }. As before, denoting E{X | X = x } by
2
2
1
1
1
1
1
2
2
µ , let us rewrite the function g(x ) as follows:
2/1 1