Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.8 IMPORTANT DISCRETE DISTRIBUTIONS

2, isr+x−^1 Cx. Therefore, the total probability of obtainingxfailures before the

rth success is

f(x)=Pr(X=x)=r+x−^1 Cxprqx,

which is called thenegative binomial distribution(see the related discussion on

p. 1137). It is straightforward to show that the MGF of this distribution is

M(t)=

(

p

1 −qet

)r

,

and that its mean and variance are given by

E[X]=

rq

p

and V[X]=

rq

p^2

.

30.8.3 The hypergeometric distribution

In subsection 30.8.1 we saw that the probability of obtainingxsuccesses inn

independenttrials was given by the binomial distribution. Suppose that thesen

‘trials’ actually consist of drawing at randomnballs, from a set ofNsuch balls

of whichMare red and the rest white. Let us consider the random variable

X= number of red balls drawn.

On the one hand, if the balls are drawnwith replacementthen the trials are

independent and the probability of drawing a red ball isp=M/Neach time.

Therefore, the probability of drawingxred balls inntrials is given by the

binomial distribution as

Pr(X=x)=

n!

x!(n−x)!

px(1−p)n−x.

On the other hand, if the balls are drawnwithout replacementthe trials are not

independent and the probability of drawing a red ball depends on how many red

balls have already been drawn. We can, however, still derive a general formula

for the probability of drawingxred balls inntrials, as follows.

The number of ways of drawingxred balls fromMisMCx, and the number

of ways of drawingn−xwhite balls fromN−MisN−MCn−x. Therefore, the

total number of ways to obtainxred balls inntrials isMCxN−MCn−x. However,

the total number of ways of drawingnobjects fromNis simplyNCn. Hence the

probability of obtainingxred balls inntrials is

Pr(X=x)=

MCxN−MCn−x

NCn

=

M!

x!(M−x)!

(N−M)!

(n−x)!(N−M−n+x)!

n!(N−n)!

N!

, (30.97)

=

(Np)!(Nq)!n!(N−n)!

x!(Np−x)!(n−x)!(Nq−n+x)!N!

, (30.98)