geometric distribution

In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:

the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or

the probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }.

Which of these one calls "the" geometric distribution is a matter of convention and convenience. If the probability of success on each trial is p, then the probability that n trials are needed to get one success is

P(X = n) = (1 - p)^{n-1}p\,

for n = 1, 2, 3, .... Equivalently the probability that there are n failures before the first success is

P(Y=n) = (1 - p)^n p\,

for n = 0, 1, 2, 3, .... In either case, the 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 X is 1/p and the variance is (1 − p)/p²;

\ E(X) = \frac{1}{p} \quad ; \quad \mbox{var}(X) = \frac{1-p}{p^2}.

Equivalently, the expected value of the geometrically distributed random variable Y is (1 − p)/p, and its variance is (1 − p)/p².

\ E(Y) = \frac{1-p}{p} \quad ; \quad \mbox{var}(Y) = \frac{1-p}{p^2}.

The probability-generating functions of X and Y are, respectively,

G_X(s) = \frac{sp}{1-s(1-p)} \quad \textrm{and} \quad G_Y(s) = \frac{p}{1-s(1-p)}, \quad |s| < (1-p)^{-1}.

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. The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y₁, ..., Y_n whose sum has the same distribution that Y has. These will not be geometrically distributed unless n = 1.

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