Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 173The formula given by Equation (6.30) is an important higher-dimensionaljoint probability distribution. It is called the multinomial distribution becauseit has the form of the general term in the multinomial expansion of(p 1 ‡ p 2 ‡‡p r ) n . We note that Equation (6.30) reduces to the binomialdistribution 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 nand p i ,we havem Xi ˆ np i ; 2 X iˆ np i …1 p i †; …6:31†and it can be shown that the covariance is given bycov…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 ina study of incomes of a given population. If, on average, 10% of the populationbelongs to the low-income group and 20% belongs to the high-income group, whatis the probability that, of the 10 persons studied, 3 will be in the low-income groupand the remaining 7 will be in the medium-income group? What is the marginaldistribution 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 10persons, X 2 be the number of medium-income persons, and X 3 be the numberof high-income persons. Then X 1 ,X 2 ,and X 3 have a multinomial distributionwith p 1 ˆ 0:1, p 2 ˆ 0:7, and p 3 ˆ 0:2; n ˆ 10.Thusp X1 X 2 X 3…3; 7; 0† ˆ 10!3!7!0! …0:1†3 …0:7† 7 …0:2† 0 0:01: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 distributionsare binomial, since X 1 ,X 2 ,..., and X r are not independent, the multinomialdistribution is not a product of binomial distributions.6.3 POISSON DISTRIBUTIONIn this section we wish to consider a distribution that is used in a wide varietyof physical situations. It is used in mathematical models for describing, in aspecific interval of time, such events as the emission of particles from aradioactive substance, passenger arrivals at an airline terminal, the distributionof dust particles reaching a certain space, car arrivals at an intersection, andmany other similar phenomena.TLFeBOOK