Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

30.15.1 The multinomial distribution

The binomial distribution describes the probability of obtainingx‘successes’ from

nindependent trials, where each trial has only two possible outcomes. This may

be generalised to the case where each trial haskpossible outcomes with respective

probabilitiesp 1 ,p 2 ,...,pk. If we consider the random variablesXi,i=1, 2 ,...,n,

to be the number of outcomes of typeiinntrials then we may calculate their

joint probability function

f(x 1 ,x 2 ,...,xk)=Pr(X 1 =x 1 ,X 2 =x 2 , ..., Xk=xk),

wherewemusthave

∑k
i=1xi=n.Inntrials the probability of obtainingx^1
outcomes of type 1, followed byx 2 outcomes of type 2 etc. is given by

px 11 px 22 ···pxkk.

However, the number of distinguishable permutations of this result is

n!

x 1 !x 2 !···xk!

,

and thus

f(x 1 ,x 2 ,...,xk)=

n!

x 1 !x 2 !···xk!

px 11 px 22 ···pxkk. (30.146)

This is themultinomial probability distribution.

Ifk= 2 then the multinomial distribution reduces to the familiar binomial

distribution. Although in this form the binomial distribution appears to be a

function of two random variables, it must be remembered that, in fact, since

p 2 =1−p 1 andx 2 =n−x 1 , the distribution ofX 1 is entirely determined by the

parameterspandn.ThatX 1 has abinomialdistribution is shown by remembering

that it represents the number of objects of a particular type obtained from

sampling with replacement, which led to the original definition of the binomial

distribution. In fact, any of the random variablesXihas a binomial distribution,

i.e. the marginal distribution of eachXiis binomial with parametersnandpi.It

immediately follows that

E[Xi]=npi and V[Xi]^2 =npi(1−pi). (30.147)

At a village fˆete patrons were invited, for a10 pentry fee, to pick without looking six

tickets from a drum containing equal large numbers of red, blue and green tickets. If five

or more of the tickets were of the same colour a prize of100 pwas awarded. A consolation

award of40 pwas made if two tickets of each colour were picked. Was a good time had by

all?

In this case, all types of outcome (red, blue and green) have the same probabilities. The
probability of obtaining any given combination of tickets is given by the multinomial
distribution withn=6,k=3andpi=^13 ,i=1, 2 ,3.