Geometric distribution - Definition and Overview

In probability theory and statistics, the geometric distribution is a discrete probability distribution -- the probability distribution of the number of Bernoulli trials needed to get one success. If the probability of success on each trial is p, then the probability that the first success is on the nth trial is:

<math>P(X = n) = (1 - p)^{n-1}p<math>

for n = 1, 2, 3, .... This sequence of probabilities is a geometric sequence.

For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set { 1, 2, 3, ... } and is a geometric distribution.

The expected value of a geometrically distributed random variable is 1/p and the variance is (1 − p)/p².

It is the special case of the negative binomial distribution in which r = 1. Like its continuous analogue (the exponential distribution), the geometric distribution is memoryless; in fact, it is the only memoryless discrete distribution.

See also negative binomial distribution.

Sometimes one defines the geometric distribution as the distribution of the number of failures before the first success, so that it is supported on the set { 0, 1, 2, 3, ... } rather than on the set { 1, 2, 3, ... }. If this is done, then the geometric distribution is infinitely divisible, i.e., if X has a geometric distribution, then for any positive integer n, there exist independent identically distributed random variables X₁, ..., X_n whose sum has the same distribution that X has. These will not be geometrically distributed unless n = 1.