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
We note that Equation (6.30) reduces to the binomial
distribution when 2 and with and
Since each Xi defined above has a binomial distribution with parameters n
and pi,wehave
and it can be shown that the covariance is given by
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-inco me persons, and X 3 be the number
of high-income persons. Then X 1 ,X 2 ,andX 3 have a multinomial distribution
with and
Thus
The marginal distribution of X 1 is binomial with and
We remark that, while the single-random-variable marginal distributions
are binomial, since X 1 ,X 2 ,..., and Xr are not independent, the multinomial
distribution is not a product of binomial distributions.
6.3 Poisson D istribution
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.
Some Important Discrete Distributions 173
(p 1 p 2 pr)n.
p 1 p,p 2 q,k 1 k, k 2 nk.
mXinpi;^2 Xinpi
1 pi;
6 : 31 cov
Xi;Xjnpipj; i;j 1 ; 2 ;...;r;i6j:
6 : 32 rp 1 0 :1,p 2 0 :7, p 3 0 :2;n10.pX 1 X 2 X 3
3 ; 7 ; 0 10!
3! 7! 0!
0 : 1 ^3 0 : 7 ^7 0 : 2 ^0 0 : 01 :
n 10 p 0 :1.