GeometricDistribution

The Geometric distribution: the number of failures before the first success, with success probability $p$ .

GeometricDistribution(p)an inert distribution object — carries

p

, unevaluated. PDF, CDF, Mean, Variance and RandomVariate all read it.

Typed signatures

Type	Package	Applies
`(real<0..1>) -> distribution`	`statistics`	always

Domain: Statistics

Details

Wolfram's own convention: support is the nonnegative integers $0, 1, 2, \ldots$ (failures before the first success), not $1, 2, \ldots$ .
$PDF(k) = p(1-p)^k$ — no support check outside the nonnegative integers (a documented divergence, same policy as unclamped PDFs elsewhere in this package).
$CDF(x) = 1 - (1-p)^{\lfloor x\rfloor + 1}$ , clamped to $0$ below $x=0$ .
$Mean = (1-p)/p$ , $Variance = (1-p)/p^2$ , both exact.
RandomVariate samples via inverse transform: $\lfloor \log U / \log(1-p) \rfloor$ .