# 11.4 - Negative Binomial Distributions

11.4 - Negative Binomial Distributions

## Example 11-1 Continued

(Are you growing weary of this example yet?) A representative from the National Football League's Marketing Division randomly selects people on a random street in Kansas City, Kansas until he finds a person who attended the last home football game. Let $$p$$, the probability that he succeeds in finding such a person, equal 0.20. Now, let $$X$$ denote the number of people he selects until he finds $$r=3$$ who attended the last home football game. What is the probability that $$X=10$$?

#### Solution

Negative Binomial Distribution

Assume Bernoulli trials — that is, (1) there are two possible outcomes, (2) the trials are independent, and (3) $$p$$, the probability of success, remains the same from trial to trial. Let $$X$$ denote the number of trials until the $$r^{th}$$ success. Then, the probability mass function of $$X$$ is:

$$f(x)=P(X=x)=\dbinom{x-1}{r-1} (1-p)^{x-r} p^r$$

for $$x=r, r+1, r+2, \ldots$$. In this case, we say that $$X$$ follows a negative binomial distribution.

NOTE!
1. There are (theoretically) an infinite number of negative binomial distributions. Any specific negative binomial distribution depends on the value of the parameter $$p$$.
2. A geometric distribution is a special case of a negative binomial distribution with $$r=1$$.

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