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

subject to the following conditions:

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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$X≔\mathrm{GeometricRandomVariable}\left(p\right)\:$

$\mathrm{ProbabilityFunction}\left(X\,u\right)$

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

$\mathrm{ProbabilityFunction}\left(X\&comma;2\right)$

$p{\left(1-p\right)}^2$

$\mathrm{Mean}\left(X\right)$

$\frac{1-p}{p}$

$\mathrm{Variance}\left(X\right)$

$\frac{1-p}{p^2}$

$Y≔\mathrm{GeometricRandomVariable}\left(\frac{1}{4}\right)\&colon;$

$\mathrm{ProbabilityFunction}\left(Y\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{CDF}\left(Y\&comma;x\right)$

$\left\{\begin{array}{cc}0& x<0\\ 1-{\left(\frac{3}{4}\right)}^{⌊x⌋+1}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{CDF}\left(Y\&comma;5\&comma;\mathrm{output}=\mathrm{plot}\right)$

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.

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

For more information on Maple 18 changes, see Updates in Maple 18.