Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYDistribution Probability law f(x) MGF E[X] V [X]binomial n C x p x q n−x (pe t + q) n np npq( ) r pnegative binomial r+x−1 C x p r q x rq1 − qe t pgeometric q x−1 pe t 1p1 − qe t phypergeometricPoisson(Np)!(Nq)!n!(N−n)!x!(Np−x)!(n−x)!(Nq−n+x)!N!λ xx! e−λ e λ(et −1)npλrqp 2qp 2N − nN − 1 npqλTable 30.1Some important discrete probability distributions.30.8 Important discrete distributionsHaving discussed some general properties of distributions, we now consider themore important discrete distributions encountered in physical applications. Theseare discussed in detail below, and summarised for convenience in table 30.1; werefer the reader to the relevant section below for an explanation of the symbolsused.30.8.1 The binomial distributionPerhaps the most important discrete probability distribution is the binomial distribution.This distribution describes processes that consist of a number of independentidentical trials with two possible outcomes, A and B = Ā. We may callthese outcomes ‘success’ and ‘failure’ respectively. If the probability of a successis Pr(A) =p then the probability of a failure is Pr(B) =q =1− p. Ifweperformn trials then the discrete random variableX = number of times A occurscan take the values 0, 1, 2,...,n; its distribution amongst these values is describedby the binomial distribution.We now calculate the probability that in n trials we obtain x successes (and son − x failures). One way of obtaining such a result is to have x successes followedby n−x failures. Since the trials are assumed independent, the probability of this ispp ···p ×} {{ }qq ···q} {{ }= p x q n−x .x times n − x timesThis is, however, just one permutation of x successes and n − x failures. The total1168