Page 127 - Probability and Statistical Inference

104 3. Multivariate Random Variables

X has the Binomial (20, 1/6) distribution, for every fixed i = 1, ..., 6. It is
i
clear, however, that must be exactly twenty in this example and hence
we conclude that X , ..., X are not all free-standing random variables. Sup-
1
6
pose that one has observed X = 2, X = 4, X = 4, X = 2 and X = 3, then X 4
5
6
3
2
1
must be 5. But, when we think of X alone, its marginal distribution is Bino-
4
mial (20, 1/6) and so by itself it can take any one of the possible values 0, 1,
2, ..., 20. On the other hand, if we assume that X = 2, X = 4, X = 4, X = 2
1
5
3
2
and X = 3, then X has to be 5. That is to say that there is certainly some kind
6
4
of dependence among these random variables X , ..., X . What is the exact
1
6
nature of this joint distribution? We discuss it next. !
We denote a vector valued random variable by X = (X , ..., X ) in general.
k
1
A discrete random variable X is said to have a multinomial distribution in-
volving n and p , ..., p , denoted by Mult (n, p , ..., p ), if and only if the joint
k
1
k
k
1
pmf of X is given by
We may think of the following experiment which gives rise to the multino-
mial pmf. Let us visualize k boxes lined up next to each other. Suppose that
we have n marbles which are tossed into these boxes in such a way that each
marble will land in one and only one of these boxes. Now, let p stand for the
i
probability of a marble landing in the i box, with 0 < p < 1, i = 1, ..., k,
th
i
. These p’s are assumed to remain same for each tossed marble.
Now, once these n marbles are tossed into these boxes, let X = the number of
i
marbles which land inside the box ≠i, i = 1, ..., k. The joint distribution of X
= (X , ..., X ) is then characterized by the pmf described in (3.2.8).
1 k
How can we verify that the expression given by (3.2.8) indeed corre-
sponds to a genuine pmf? We must check the requirements stated in (3.2.2).
The first requirement certainly holds. In order to verify the requirement (ii),
we need an extension of the Binomial Theorem. For completeness, we state
and prove the following result.
Theorem 3.2.1 (Multinomial Theorem) Let a , ..., a be arbitrary real
1
k
numbers and n be a positive integer. Then, we have