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Operations Research Models and Methods
Computation Section
Subunit Discrete Distributions
 - Negative Binomial

For the negative binomial distribution the random variable is the number of failures before the rth success is observed. The distribution has two parameters. The parameter p is the probability of success on any one trial and the parameter r is the number of successes to be observed before the experiment is complete.

The geometric distribution is a special case with r equal to 1.


Example: : In the game of craps, you decide to play until you lose 5 games. You wonder how many games you will play with this termination rule. The probability of losing any one game is 0.5071. The games are a series of independent Bernoulli trials, and the random variable is the number of wins until the fifth loss. This is a situation described by the negative binomial distribution.

For the example, we perversely defined success as a “loss” with p the probability of a success equal to 0.5071 for the example. The random variable is the number of trials that result in 0 before the rth 1 is observed. For this case, r = 5.

It is important to remember that the random variable is not the total number of trials but the number of failed trials before the rth success. In the solution, the entry for 0 describes the probability that the first five plays were losses and there were no wins.


An alternative statement of the random variable for the negative binomial distribution is the number of trials until the rth success. The count includes the successes as well as the failures. This random variable must be at least r.
The figure shows the example modeled with the alternative statement. The distribution is shifted by 5. The mean is increased by 5, but the variance and the other moments remain the same.
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Operations Research Models and Methods
by Paul A. Jensen
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