11.1 - Geometric Distributions11.1 - Geometric Distributions
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 \(X\)?
- 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:
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 \(p\).