Page 187 - Fundamentals of Probability and Statistics for Engineers

170 Fundamentals of Probability and Statistics for Engineers

and P(B) is simply
P B†ˆ p: 6:20†

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

k 1 r k r
p k†ˆ p q ; k ˆ r; r ‡ 1; ... : 6:21†
X
r 1
We note that, as expected, it reduces to the geometric distribution when r ˆ 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).
A useful variant of this distribution is obtained if we let Y ˆ X r. The ran-
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
the rth success.
The probability mass function of Y, p (m), is obtained from Equation (6.21)
Y
upon replacing k by m ‡ r. Thus,

m ‡ r 1 r m
p m†ˆ p q
Y
r 1
6:22†

m ‡ r 1
r m
ˆ p q ; m ˆ 0; 1; 2; ... :
m
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.
Recalling a more general definition of the binomial coefficient
a a a 1† ... a j ‡ 1†

ˆ ; 6:23†
j j!
for any real a and any positive integer j, direct evaluation shows that the
binomial coefficient in Equation (6.22) can be written in the form

m ‡ r 1 m r

ˆ 1† : 6:24†
m m
Hence,

r m

r
p m†ˆ p q† ; m ˆ 0; 1; 2; ... ; 6:25†
Y
m

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