Page 147 - Probability and Statistical Inference

124 3. Multivariate Random Variables

Next, we summarize some more useful results for algebraic manipulations
involving expectations, variances, and covariances.
Theorem 3.4.3 Let us write a , ..., a and b , ..., b for arbitrary but fixed
k
1
k
1
real numbers. Also recall the arbitrary k-dimensional random variable X =
(X , ..., X ) which may be discrete or continuous. Then, we have
1 k

Proof The parts (i) and (ii) follow immediately from the Theorem 3.3.2
and (3.4.3). The parts (iii) and (iv) follow by successively applying the bilin-
ear property of the covariance operation stated precisely in the Theorem 3.4.1,
part (ii). The details are left out as the Exercise 3.4.5. ¢

One will find an immediate application of Theorem 3.4.3
in a multinomial distribution.

3.4.1 The Multinomial Case

Recall the multinomial distribution introduced in the Section 3.2.2. Sup-
pose that X = (X , ..., X ) has the Mult (n, p , ..., p ) distribution with the
k
k
1
k
1
associated pmf given by (3.2.8). We had mentioned that X has the Binomial(n,
i
p ) distribution for all fixed i = 1, ..., k. How should we proceed to derive the
i
expression of the covariance between X and X for all fixed i ≠ j = 1, ..., k?
i j
Recall the way we had motivated the multinomial pmf given by (3.2.8).
Suppose that we have n marbles and these are tossed so that each marble
lands in one of the k boxes. The probability that a marble lands in box #i is,
say, p , i = 1, ..., k, Then, let X be the number of marbles land-
i
l
ing in the box #l, l = i, j. Define

Then, utilizing (3.4.14) we can write