# 11.1 - Geometric Distributions

11.1 - Geometric Distributions

## Example 11-1

A representative from the National Football League's Marketing Division randomly selects people on a random street in Kansas City, Missouri 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. And, let $$X$$ denote the number of people he selects until he finds his first success. What is the probability mass function of $$X$$?

#### Solution

Geometric 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 first success. Then, the probability mass function of $$X$$ is:

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

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

Note that there are (theoretically) an infinite number of geometric distributions. Any specific geometric distribution depends on the value of the parameter $$p$$.

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