Page 128 - Probability and Statistical Inference

3. Multivariate Random Variables 105

Proof First consider the case k = 2. By the Binomial Theorem, we get

which is the same as (3.2.9). Now, in the case k = 3, by using (3.2.10)
repeatedly let us write

which is the same as (3.2.9). Next, let us assume that the expansion (3.2.9)
holds for all n ≤r, and let us prove that the same expansion will then hold for
n = r + 1. By mathematical induction, the proof will then be complete. Let us
write

which is the same as (3.2.9). The proof is now complete. ¢

The fact that the expression in (3.2.8) satisfies the requirement (ii) stated
in (3.2.2) follows immediately from the Multinomial Theorem. That is, the
function f(x) given by (3.2.8) is a genuine multivariate pmf.

The marginal distribution of the random variable X is
i
Binomial (n,p ) for each fixed i = 1, ..., k.
i
In the case of the multinomial distribution (3.2.8), suppose that we focus on
the random variable X alone for some arbitrary but otherwise fixed
i