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Statistics[Distributions]

 Geometric
 geometric distribution

 Calling Sequence Geometric(p) GeometricDistribution(p)

Parameters

 p - probability of success

Description

 • The geometric distribution is a discrete probability distribution with probability function given by:

$f\left(t\right)=\mathrm{piecewise}\left(t<0,0,p{\left(1-p\right)}^{t}\right)$

 subject to the following conditions:

$0

 • The geometric distribution has the lack of memory property: the probability of an event occurring in the next time interval of an exponential distribution is independent of the amount of time that has already passed.
 • The geometric variate is a special case of the NegativeBinomial variate with number of trials parameter $x=1$.
 • The continuous analog of the geometric variate is the Exponential variate.
 • Note that the Geometric command is inert and should be used in combination with the RandomVariable command.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Geometric}\left(p\right)\right):$
 > $\mathrm{ProbabilityFunction}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ {p}{}{\left({1}{-}{p}\right)}^{{u}}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{ProbabilityFunction}\left(X,2\right)$
 ${p}{}{\left({1}{-}{p}\right)}^{{2}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{1}{-}{p}}{{p}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{1}{-}{p}}{{{p}}^{{2}}}$ (4)

References

 Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
 Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.
 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

 See Also