Applied Statistics and Probability for Engineers

3-7 GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS 81

Because at least rtrials are required to obtain rsuccesses, the range of Xis from rto. In the

special case that r1, 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

Xdenote the total number of trials required to obtain rsuccesses. Let denote the number of

trials required to obtain the first success, let denote the number of extra trials required to

obtain the second success, let 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

. Because of the lack of memory property, each of the random vari-
ables has a geometric distribution with the same value of p. Consequently, a
negative binomial random variable can be interpreted as the sum of rgeometric 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 1 , X 2 ,p, Xr

XX 1    X 2     p Xr

X 3

X 2

X 1

0

0.02

0.04

0.08

0.12

f (x)

5

0.06

0.10

0

5

20 40 60 80 100 120

x

10

p

0.1

0.4

0.4

Figure 3-10 Negative

binomial distributions

for selected values of the

parameters rand p.

1 2 3 4 5 6 7 8 9 10 11 12

Trials

indicates a trial that results in a "success".

X 1 X 2 X 3

X = X 1 + X 2 + X 3

Figure 3-11 Negative

binomial random

variable represented as

a sum of geometric

random variables.

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