Page 190 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 173

The formula given by Equation (6.30) is an important higher-dimensional
joint probability distribution. It is called the multinomial distribution because
it has the form of the general term in the multinomial expansion of
n
(p 1 ‡ p 2 ‡ ‡ p r ) . We note that Equation (6.30) reduces to the binomial
distribution when r ˆ 2 and with p 1 ˆ p, p 2 ˆ q, k 1 ˆ k, and k 2 ˆ n k.
Since each X i defined above has a binomial distribution with parameters n
and p i , we have

ˆ np i ; 2 ˆ np i 1 p i †; 6:31†
m X i
X i
and it can be shown that the covariance is given by
cov X i ; X j †ˆ np i p j ; i; j ˆ 1; 2; ... ; r; i 6ˆ j: 6:32†

Example 6.10. Problem: income levels are classified as low, medium, and high in
a study of incomes of a given population. If, on average, 10% of the population
belongs to the low-income group and 20% belongs to the high-income group, what
is the probability that, of the 10 persons studied, 3 will be in the low-income group
and the remaining 7 will be in the medium-income group? What is the marginal
distribution of the number of persons (out of 10) at the low-income level?
Answer: let X 1 be the number of low-income persons in the group of 10
persons, X 2 be the number of medium-income persons, and X 3 be the number
of high-income persons. Then X 1 , X 2 , and X 3 have a multinomial distribution
with p 1 ˆ 0:1, p 2 ˆ 0:7, and p 3 ˆ 0:2; n ˆ 10.
Thus

10! 3 7 0
p 3; 7; 0†ˆ 0:1† 0:7† 0:2†  0:01:
3!7!0!
X 1 X 2 X 3
The marginal distribution of X 1 is binomial with n ˆ 10 and p ˆ 0:1.
We remark that, while the single-random-variable marginal distributions
are binomial, since X 1 , X 2 ,.. ., and X r are not independent, the multinomial
distribution is not a product of binomial distributions.

6.3 POISSON DISTRIBUTION

In this section we wish to consider a distribution that is used in a wide variety
of physical situations. It is used in mathematical models for describing, in a
specific interval of time, such events as the emission of particles from a
radioactive substance, passenger arrivals at an airline terminal, the distribution
of dust particles reaching a certain space, car arrivals at an intersection, and
many other similar phenomena.

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