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

subject to the following conditions:

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.

There are two different but intimately related distributions that are frequently called geometric distributions, one of which is the distribution described on this help page. Let us call this the Maple definition. It is often defined by saying it describes the number of failures before the first success in a sequence of independent Bernoulli distributed trials. The other distribution often called the geometric distribution can then be defined as the number of trials up to and including the first success; let us call it the alternative definition.  If $X$ is a random variable with the geometric distribution according to the Maple definition, then $X+1$ is a random variable with the geometric distribution (with the same value for the parameter p) according to the alternative definition. The alternative definition yields the following probability function:

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

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

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

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

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

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

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

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

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

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

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.