Fundamentals of Probability and Statistics for Engineers

and P(B) is simply

Substituting Equations (6.19) and (6.20) into Equation (6.18) results in

We note that, as expected, it reduces to the geometric distribution when r 1.
The distribution defined by Equation (6. 21 ) is known as the negative binomial,
or Pascal, distribution with parameters r and p. It is often denoted by NB(r,p).
Ausefulvariantofthisdistributionisobtainedifwelet Theran-
dom variable Y is the number of Bernoulli trials beyond r needed for the realiza-
tion of the rth success, or it can be interpreted as the number of failures before
therthsuccess.
Theprobabilitymassfunctionof isobtainedfromEquation( 6. 21 )
upon replacing k by m r. Thus,

We see that random variable Y has the convenient property that the range of
m begins at zero rather than r for values associated with X.
Recallingamoregeneraldefinitionofthebinomialcoefficient

for any real a and any positive integer j, direct evaluation shows that the
binomialcoefficientinEquation( 6. 22 )canbewrittenintheform

Hence,

170 FundamentalsofProbabilityandStatisticsforEngineers

P…B†ˆp: … 6 : 20 †

pX…k†ˆ

k 1

r 1



prqkr; kˆr;r‡ 1 ;...: … 6 : 21 †

ˆ

YˆXr.

Y,pY(m),

‡

pY…m†ˆ

m‡r 1

r 1



prqm

ˆ

m‡r 1

m



prqm; mˆ 0 ; 1 ; 2 ;...:

… 6 : 22 †

a

j



ˆ

a…a 1 †...…aj‡ 1 †

j!

; … 6 : 23 †

m‡r 1

m



ˆ… 1 †m

r

m



: … 6 : 24 †

pY…m†ˆ

r

m



pr…q†m; mˆ 0 ; 1 ; 2 ;...; … 6 : 25 †