Geometric distribution: Definition from Answers.com

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, ...}

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.

Geometric
Probability mass function
Cumulative distribution function
Parameters	$0< p \leq 1$ success probability (real)	$0< p \leq 1$ success probability (real)
Support	$k \in \{1,2,3,\dots\}\!$	$k \in \{0,1,2,3,\dots\}\!$
Probability mass function (pmf)	$(1 - p)^{k-1}\,p\!$	$(1 - p)^{k}\,p\!$
Cumulative distribution function (cdf)	$1-(1 - p)^k\!$	$1-(1 - p)^{k+1}\!$
Mean	$\frac{1}{p}\!$	$\frac{1-p}{p}\!$
Median	$\left\lceil \frac{-\log(2)}{\log(1-p)} \right\rceil\!$ (not unique if $- log(2) / log(1 - p)$ is an integer)
Mode	1	0
Variance	$\frac{1-p}{p^2}\!$	$\frac{1-p}{p^2}\!$
Skewness	$\frac{2-p}{\sqrt{1-p}}\!$	$\frac{2-p}{\sqrt{1-p}}\!$
Excess kurtosis	$6+\frac{p^2}{1-p}\!$	$6+\frac{p^2}{1-p}\!$
Entropy	$\frac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}\!$
Moment-generating function (mgf)	$\frac{pe^t}{1-(1-p) e^t}\!$	$\frac{p}{1-(1-p)e^t}\!$
Characteristic function	$\frac{pe^{it}}{1-(1-p)\,e^{it}}\!$	$\frac{p}{1-(1-p)\,e^{it}}\!$

These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of the number X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the range explicitly.

If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is

$\Pr(X = k) = (1 - p)^{k-1}\,p\,$

for k = 1, 2, 3, ....

Equivalently, if the probability of success on each trial is p, then the probability that there are k failures before the first success is

$\Pr(Y=k) = (1 - p)^k\,p\,$

for k = 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.

1 Moments and cumulants
2 Parameter estimation
3 Other properties
4 Related distributions
5 See also
6 External links

Moments and cumulants

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

$\mathrm{E}(X) = \frac{1}{p}, \qquad\mathrm{var}(X) = \frac{1-p}{p^2}.$

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

$\mathrm{E}(Y) = \frac{1-p}{p}, \qquad\mathrm{var}(Y) = \frac{1-p}{p^2}.$

Let μ = (1 − p)/p be the expected value of Y. Then the cumulants $κ n$ of the probability distribution of Y satisfy the recursion

$\kappa_{n+1} = \mu(\mu+1) \frac{d\kappa_n}{d\mu}.$

Outline of proof: That the expected value is (1 − p)/p can be shown (why summation and integration is exchanged is never shown) in the following way. Let Y be as above. Then

$\begin{align} \mathrm{E}(Y) & {} =\sum_{k=0}^\infty (1-p)^k p\cdot k \\ & {} =p\sum_{k=0}^\infty(1-p)^k k \\ & {} = p\left[\frac{d}{dp}\left(-\sum_{k=0}^\infty (1-p)^k\right)\right](1-p) \\ & {} =-p(1-p)\frac{d}{dp}\frac{1}{p}=\frac{1-p}{p}. \end{align}$

Parameter estimation

For both variants of the geometric distribution, the parameter p can be estimated by equating the expected value with the sample mean. This is the method of moments, which in this case happens to yield maximum likelihood estimates of p.

Specifically, for the first variant let $k_1,\dots,k_n$ be a sample where $k_i \geq 1$ for $i=1,\dots,n$ . Then p can be estimated as

$\widehat{p} = \left(\frac1n \sum_{i=1}^n k_i\right)^{-1}. \!$

In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta(α, β) prior, then the posterior distribution is

$p \sim \mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^n (k_i-1)\right). \!$

The posterior mean $E[p]$ approaches the maximum likelihood estimate $\widehat{p}$ as α and β approach zero.

In the alternative case, let $k_1,\dots,k_n$ be a sample where $k_i \geq 0$ for $i=1,\dots,n$ . Then p can be estimated as

$\widehat{p} = \left(1 + \frac1n \sum_{i=1}^n k_i\right)^{-1}. \!$

The posterior distribution of p given a Beta(α, β) prior is

$p \sim \mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^n k_i\right). \!$

Again the posterior mean $E[p]$ approaches the maximum likelihood estimate $\widehat{p}$ as α and β approach zero.

Other properties

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

$G_X(s) = \frac{s\,p}{1-s\,(1-p)}, \!$

$G_Y(s) = \frac{p}{1-s\,(1-p)}, \quad |s| < (1-p)^{-1}. \!$

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; they follow a negative binomial distribution.

The decimal digits of the geometrically distributed random variable Y are a sequence of independent (and not identically distributed) random variables. For example, the hundreds digit D has this probability distribution:

$\Pr(D=d) = {q^{100d} \over {1 + q^{100} + q^{200} + \cdots + q^{900}}},$

where q = 1 − p, and similarly for the other digits, and, more generally, similarly for numeral systems with other bases than 10. When the base is 2, this shows that a geometrically distributed random variable can be written as a sum of independent random variables whose probability distributions are indecomposable.

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

$Z = \sum_{m=1}^r Y_m$

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

$W = \min_{m} Y_m\,$

is also geometrically distributed, with parameter p given by

1 −	∏	(1 − p^(m)).
	m

Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable X_k has a Poisson distribution with expected value r^k/k. Then

$\sum_{k=1}^\infty k\,X_k$

has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value r/(1 − r).

The exponential distribution is the continuous analogue of the geometric distribution. If X is an exponentially distributed random variable with parameter λ, then

$Y = \lfloor X \rfloor$ (where $\lfloor \rfloor$ is the floor (or greatest integer) function)

is a geometrically distributed random variable with parameter

p = 1 - e - λ

and taking values in the set {0, 1, 2, ...}. This can be used to generate geometrically distributed pseudorandom numbers by first generating exponentially distributed pseudorandom numbers from a uniform pseudorandom number generator.

External links

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway–Maxwell–Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell–Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

v • d • e Some common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • Laplace • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

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Geometric distribution

Contents

Moments and cumulants

Parameter estimation

Other properties

Related distributions

See also

External links