Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITY30.15.1 The multinomial distributionThe binomial distribution describes the probability of obtaining x ‘successes’ fromn independent trials, where each trial has only two possible outcomes. This maybe generalised to the case where each trial has k possible outcomes with respectiveprobabilities p 1 , p 2 , ..., p k . If we consider the random variables X i , i =1, 2,...,n,to be the number of outcomes of type i in n trials then we may calculate theirjoint probability functionf(x 1 ,x 2 ,...,x k )=Pr(X 1 = x 1 ,X 2 = x 2 , ..., X k = x k ),wherewemusthave ∑ ki=1 x i = n. Inn trials the probability of obtaining x 1outcomes of type 1, followed by x 2 outcomes of type 2 etc. is given byp x11 px2 2 ···pxk k .However, the number of distinguishable permutations of this result isn!x 1 !x 2 ! ···x k ! ,and thusn!f(x 1 ,x 2 ,...,x k )=x 1 !x 2 ! ···x k ! px1 1 px2 2 ···pxk k . (30.146)This is the multinomial probability distribution.If k = 2 then the multinomial distribution reduces to the familiar binomialdistribution. Although in this form the binomial distribution appears to be afunction of two random variables, it must be remembered that, in fact, sincep 2 =1− p 1 and x 2 = n − x 1 , the distribution of X 1 is entirely determined by theparameters p and n. ThatX 1 has a binomial distribution is shown by rememberingthat it represents the number of objects of a particular type obtained fromsampling with replacement, which led to the original definition of the binomialdistribution. In fact, any of the random variables X i has a binomial distribution,i.e. the marginal distribution of each X i is binomial with parameters n and p i .Itimmediately follows thatE[X i ]=np i and V [X i ] 2 = np i (1 − p i ). (30.147)◮At a village fête patrons were invited, for a 10 p entry fee, to pick without looking sixtickets from a drum containing equal large numbers of red, blue and green tickets. If fiveor more of the tickets were of the same colour a prize of 100 p was awarded. A consolationaward of 40 p was made if two tickets of each colour were picked. Was a good time had byall?In this case, all types of outcome (red, blue and green) have the same probabilities. Theprobability of obtaining any given combination of tickets is given by the multinomialdistribution with n =6,k =3andp i = 1 , i =1, 2, 3.31208