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154 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

5-2.2 Multinomial Probability Distribution

A joint probability distribution for multiple discrete random variables that is quite useful is an
extension of the binomial. The random experiment that generates the probability distribution
consists of a series of independent trials. However, the results from each trial can be catego-
rized into one of k classes.

EXAMPLE 5-12 We might be interested in a probability such as the following. Of the 20 bits received, what is
the probability that 14 are excellent, 3 are good, 2 are fair, and 1 is poor? Assume that the clas-
sifications of individual bits are independent events and that the probabilities of E, G, F, and
P are 0.6, 0.3, 0.08, and 0.02, respectively. One sequence of 20 bits that produces the speci-
fied numbers of bits in each class can be represented as

EEEEEEEEEEEEEEGGGFFP

Using independence, we find that the probability of this sequence is
14
3
1
2
P1EEEEEEEEEEEEEEGGGFFP2 0.6 0.3 0.08 0.02 2.708 10 9
Clearly, all sequences that consist of the same numbers of E’s, G’s, F’s, and P’s have the same
probability. Consequently, the requested probability can be found by multiplying 2.708
10 9 by the number of sequences with 14 E’s, three G’s, two F’s, and one P. The number of
sequences is found from the CD material for Chapter 2 to be

20!
2325600
14!3!2!1!

Therefore, the requested probability is
, , ,
9
P114E s, three G s, two F s, and one P2 232560012.708 10 2 0.0063
Example 5-12 leads to the following generalization of a binomial experiment and a bino-
mial distribution.

Multinomial
Distribution Suppose a random experiment consists of a series of n trials. Assume that

(1) The result of each trial is classified into one of k classes.
(2) The probability of a trial generating a result in class 1, class 2, p , class k
, p , p , p , respectively.
is constant over the trials and equal to p 1 2 k
(3) The trials are independent.
The random variables X , X , p , X that denote the number of trials that result in
2
k
1
class 1, class 2, p , class k, respectively, have a multinomial distribution and the
joint probability mass function is
n!
P1X x , X x , p , X x 2 p p p p x k (5-13)
x 1
x 2
1
k
1
2
1
k
k
2
2
k
x 1 !x 2 ! p x !
for x x p x n and p p p p . 1
1
1
k
k
2
2