Applied Statistics and Probability for Engineers

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 kclasses.

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

Pare 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

Using independence, we find that the probability of this sequence is

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

Therefore, the requested probability is

Example 5-12 leads to the following generalization of a binomial experiment and a bino-

mial distribution.

P 114 E

,

s, three G

,

s, two F

,

s, and one P 2  23256001 2.708 10 ^92 0.0063

20!

14! 3! 2! 1!

 2325600

P 1 EEEEEEEEEEEEEEGGGFFP 2 0.6^14 0.3^3 0.08^2 0.02^1 2.708 10 ^9

EEEEEEEEEEEEEEGGGFFP

Multinomial

Distribution Suppose a random experiment consists of a series of ntrials. Assume that

(1) The result of each trial is classified into one of kclasses.

(2) The probability of a trial generating a result in class 1, class 2, , class k

is constant over the trials and equal to p 1 , p 2 ,, pk, respectively.

(3) The trials are independent.

The random variables X 1 , X 2 ,, Xkthat denote the number of trials that result in

class 1, class 2, , class k, respectively, have a multinomial distributionand the

joint probability mass function is

(5-13)

for and .x 1 x 2 pxkn p 1 p 2 ppk 1

P 1 X 1 x 1 , X 2 x 2 ,p, Xkxk 2 

n!

x 1 !x 2! p xk!

p 1 x^1 p 2 x^2 ppxkk

p

p

p

p

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