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Student[Statistics]

 GeometricRandomVariable
 geometric random variable

 Calling Sequence GeometricRandomVariable(p)

Parameters

 p - probability of success

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ p{\left(1-p\right)}^{t}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0

 • The geometric random variable has the lack of memory property: the probability of an event occurring in the next time interval of an exponential random variable 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 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 $p{\left(1-p\right)}^{t-1}$ in the probability function.

Examples

 > $\mathrm{with}\left({\mathrm{Student}}_{\mathrm{Statistics}}\right):$
 > $X≔\mathrm{GeometricRandomVariable}\left(p\right):$
 > $\mathrm{ProbabilityFunction}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ {p}{}{\left({1}{-}{p}\right)}^{{u}}& {\mathrm{otherwise}}\end{array}\right\$ (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)
 > $Y≔\mathrm{GeometricRandomVariable}\left(\frac{1}{4}\right):$
 > $\mathrm{ProbabilityFunction}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$ > $\mathrm{CDF}\left(Y,x\right)$
 $\left\{\begin{array}{cc}{0}& {x}{<}{0}\\ {1}{-}{\left(\frac{{3}}{{4}}\right)}^{⌊{x}⌋{+}{1}}& {\mathrm{otherwise}}\end{array}\right\$ (5)
 > $\mathrm{CDF}\left(Y,5,\mathrm{output}=\mathrm{plot}\right)$ 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.

Compatibility

 • The Student[Statistics][GeometricRandomVariable] command was introduced in Maple 18.