# GeometricDistribution

represents a geometric distribution with probability parameter p.

# Details • The probability for value in a geometric distribution is for non-negative integers, and is zero otherwise. »
• is the distribution of the number of failures in a sequence of trials with success probability p before a success occurs.
• GeometricDistribution allows p to be a dimensionless quantity. »
• GeometricDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

# Background & Context

• represents a discrete statistical distribution defined at integer values and parametrized by a non-negative real number . The geometric distribution has a discrete probability density function (PDF) that is monotonically decreasing, with the parameter p determining the height and steepness of the PDF. The geometric distribution is sometimes referred to as the Furry distribution.
• The geometric distribution is sometimes said to be the discrete analog of the exponential distribution (ExponentialDistribution). It can be defined as the distribution that models the number of Bernoulli trials (i.e. number of trials of a variate having a BernoulliDistribution) needed to obtain a single success. The geometric distribution has been used to model a number of different phenomena across many fields, including the behavior of competing plant populations, dynamics of ticket control, particulars of congenital malformations, and estimation of animal abundance. Moreover, the geometric distribution has been used widely in reliability theory and is a staple in Markov chain models for meteorology, queueing theory, and applied stochastics.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a geometric distribution. Distributed[x,GeometricDistribution[p]], written more concisely as xGeometricDistribution[p], can be used to assert that a random variable x is distributed according to a geometric distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[GeometricDistribution[p],x] and CDF[GeometricDistribution[p],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
• DistributionFitTest can be used to test if a given dataset is consistent with a geometric distribution, EstimatedDistribution to estimate a geometric parametric distribution from given data, and FindDistributionParameters to fit data to a geometric distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic geometric distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic geometric distribution.
• TransformedDistribution can be used to represent a transformed geometric distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a geometric distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving geometric distributions.
• GeometricDistribution is related to a number of other statistical distributions. For example, GeometricDistribution is a special case of the more general NegativeBinomialDistribution, in the sense that has precisely the same PDF as and that the sum of is distributed according to , provided that XiGeometricDistribution[p] for all . GeometricDistribution is also a transformation of PascalDistribution and can be viewed as a parameter mixture of PoissonDistribution with either GammaDistribution or ExponentialDistribution, in the sense that both ParameterMixtureDistribution[PoissonDistribution[μ],μGammaDistribution[1,(1-p)/p]] and are the same as . GeometricDistribution is also related to WaringYuleDistribution, BinomialDistribution, PascalDistribution, and HypergeometricDistribution.

# Examples

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## Basic Examples(3)

Probability mass function:

 In:= Out= In:= Out= Cumulative distribution function:

 In:= Out= In:= Out= Mean and variance of a geometric distribution:

 In:= Out= In:= Out= ## Possible Issues(2)

Introduced in 2007
(6.0)
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Updated in 2016
(10.4)