Geometric distribution

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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

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

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

<math>P(Y=n) = (1 - p)^n p\,</math>

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 with p = 1/6.

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

<math>\ E(X) = \frac{1}{p}, \quad \mbox{var}(X) = \frac{1-p}{p^2}.</math>

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

<math>\ E(Y) = \frac{1-p}{p},\quad \mbox{var}(Y) = \frac{1-p}{p^2}.</math>

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

<math>G_X(s) = \frac{sp}{1-s(1-p)} \quad</math>

<math>G_Y(s) = \frac{p}{1-s(1-p)}, \quad |s| < (1-p)^{-1}.</math>

Like its continuous analogue (the exponential distribution), the geometric distribution is memoryless. That means that if you intend to repeat an experiment until the first success, then, given that the first success has not yet occurred, the conditional probability distribution of the number of additional trials does not depend on how many failures have been observed. The die one throws or the coin one tosses does not have a "memory" of these failures. The geometric distribution is in fact the only memoryless discrete distribution.

Among all discrete probability distributions supported on {1, 2, 3, ... } with given expected value μ, the geometric distribution X with parameter p = 1/μ is the one with the largest entropy.

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.

Related distributions

The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y₁,...,Y_r are independent geometrically distributed variables with parameter p, then <math>Z = \sum_{m=1}^r Y_m</math> follows a negative binomial distribution with parameters r and p.

If Y₁,...,Y_r are independent geometrically distributed variables (with possibly different success parameters p_m), then their minimum <math>W = \min_{m} Y_m</math> is also geometrically distributed, with parameter p given by <math>1-\prod_{m}(1-p_m)</math>.

External links

• Template:Planetmath reference
• Geometric distribution on MathWorld.de:Geometrische Verteilung

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