Statistics[Distributions]
Geometric
geometric distribution
Calling Sequence
Parameters
Description
Examples
References
Geometric(p)
GeometricDistribution(p)
p

probability of success
The geometric distribution is a discrete probability distribution with probability function given by:
$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ p{\left(1p\right)}^{t}& \mathrm{otherwise}\end{array}\right.$
subject to the following conditions:
$0<p,p\le 1$
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.
Note that the distribution above is for the number of failures $\mathrm{before}$ the first success. The other common convention is for the number of trials with the last being the first success. That is, the other convention would have $p{\left(1p\right)}^{t1}$ in the probability function.
$\mathrm{with}\left(\mathrm{Statistics}\right)\:$
$X\u2254\mathrm{RandomVariable}\left(\mathrm{Geometric}\left(p\right)\right)\:$
$\mathrm{ProbabilityFunction}\left(X\,u\right)$
$\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ {p}{}{\left({1}{}{p}\right)}^{{u}}& {\mathrm{otherwise}}\end{array}\right.$
$\mathrm{ProbabilityFunction}\left(X\,2\right)$
${p}{}{\left({1}{}{p}\right)}^{{2}}$
$\mathrm{Mean}\left(X\right)$
$\frac{{1}{}{p}}{{p}}$
$\mathrm{Variance}\left(X\right)$
$\frac{{1}{}{p}}{{{p}}^{{2}}}$
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
Statistics
Statistics[RandomVariable]
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