Page 129 - Probability and Statistical Inference

106 3. Multivariate Random Variables

i = 1, ..., k. The question is this: Out of the n marbles, how many (that is, X)
i
would land in the i box? We are simply counting how many marbles would fall
th
in the i box and how many would fall outside, that is in any one of the other k
th
– 1 boxes. This is the typical binomial situation and hence we observe that the
random variable X has the Binomial(n, p) distribution for each fixed i = 1, ..., k.
i
i
Hence, from our discussions on the binomial distribution in Section 2.2.2 and
(2.2.17), it immediately follows that

Example 3.2.7 (Example 3.2.6 Continued) In the die rolling example, X =
(X , ..., X ) has the Mult (n, p , ..., p ) distribution where n = 20, k = 6, and p 1
1
k
1
k
k
= ... = p = 1/6. !
6
The derivation of the moment generating function (mgf) of the
multinomial distribution and some of its applications are
highlighted in Exercise 3.2.8.
The following theorems are fairly straightforward to prove. We leave their
proofs as the Exercises 3.2.5-3.2.6.
Theorem 3.2.2 Suppose that the random vector X = (X , ..., X ) has the
1
k
Mult (n, p , ..., p ) distribution. Then, any subset of the X variables of size
k 1 k
r, namely (X , ..., X ) has a multinomial distribution in the sense that (X
i1 ir i1, ...,
X , ) is Mult (n, p , ..., p , ) where 1 ≤ i < i < ... < i ≤ k
ir r+1 i1 ir 1 2 r
and .
Theorem 3.2.3 Suppose that the random vector X = (X , ..., X ) has the
k
1
Mult (n, p , ..., p ) distribution. Consider any subset of the X variables of
k
1
k
size r, namely (X , ..., X ). The conditional joint distribution of (X , ..., X )
i1
ir
i1
ir
given all the remaining X’s is also multinomial with its conditional pmf

Example 3.2.8 (Example 3.2.7 Continued) In the die rolling example,
suppose that we are simply interested in counting how many times the faces
with the numbers 1 and 5 land up. In other words, our focus is on the three