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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 81

3-7 GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS 81

0.12
p
5 0.1
0.10 5 0.4
10 0.4

0.08

0.06
f (x)

0.04

0.02

Figure 3-10 Negative
binomial distributions 0
for selected values of the 0 20 40 60 80 100 120
parameters r and p. x

Because at least r trials are required to obtain r successes, the range of X is from r to . In the

special case that r 1, a negative binomial random variable is a geometric random variable.
Selected negative binomial distributions are illustrated in Fig. 3-10.
The lack of memory property of a geometric random variable implies the following. Let
X denote the total number of trials required to obtain r successes. Let X 1 denote the number of
trials required to obtain the first success, let X 2 denote the number of extra trials required to
obtain the second success, let X 3 denote the number of extra trials to obtain the third success,
and so forth. Then, the total number of trials required to obtain r successes is
X X X p X r . Because of the lack of memory property, each of the random vari-
1
2
ables X , X , p , X r has a geometric distribution with the same value of p. Consequently, a
2
1
negative binomial random variable can be interpreted as the sum of r geometric random vari-
ables. This concept is illustrated in Fig. 3-11.
Recall that a binomial random variable is a count of the number of successes in n
Bernoulli trials. That is, the number of trials is predetermined, and the number of successes is
random. A negative binomial random variable is a count of the number of trials required to

X = X + X + X 3
2
1
X X X
Figure 3-11 Negative 1 2 3
binomial random * * *
variable represented as 1 2 3 4 5 6 7 8 9 10 11 12
a sum of geometric Trials
random variables. * indicates a trial that results in a "success".