Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

Distribution Probability lawf(x)MGFE[X] V[X]

binomial nCxpxqn−x (pet+q)n np npq

negative binomial r+x−^1 Cxprqx

(

p

1 −qet

)r

rq

p

rq

p^2

geometric qx−^1 p

pet

1 −qet

1

p

q

p^2

hypergeometric x!(Np(−Npx)!()!(nNq−x)!)!(n!(NqN−−nn)!+x)!N! np

N−n

N− 1

npq

Poisson

λx

x!

e−λ eλ(e

t−1)

λλ

Table 30.1 Some important discrete probability distributions.

30.8 Important discrete distributions

Having discussed some general properties of distributions, we now consider the

more important discrete distributions encountered in physical applications. These

are discussed in detail below, and summarised for convenience in table 30.1; we

refer the reader to the relevant section below for an explanation of the symbols

used.

30.8.1 The binomial distribution

Perhaps the most important discrete probability distribution is thebinomial dis-

tribution. This distribution describes processes that consist of a number of inde-

pendent identicaltrialswith two possible outcomes,AandB=A ̄. We may call

these outcomes ‘success’ and ‘failure’ respectively. If the probability of a success

is Pr(A)=pthen the probability of a failure is Pr(B)=q=1−p.Ifweperform

ntrials then the discrete random variable

X= number of timesAoccurs

can take the values 0, 1 , 2 ,...,n; its distribution amongst these values is described

by thebinomial distribution.

We now calculate the probability that inntrials we obtainxsuccesses (and so

n−xfailures). One way of obtaining such a result is to havexsuccesses followed

byn−xfailures. Since the trials are assumed independent, the probability of this is

pp···p

︸︷︷ ︸

xtimes

× qq···q

︸ ︷︷︸

n−xtimes

=pxqn−x.

This is, however, just one permutation ofxsuccesses andn−xfailures. The total