GammaDistribution

The Gamma distribution with shape $k$ and scale $\theta$ .

GammaDistribution(k, theta)an inert distribution object with shape

k

and scale

\theta

GammaDistribution(k)scale defaults to

1

. NOT a Wolfram call form — GammaDistribution[alpha] alone errors there (GammaDistribution::argbu, confirmed against wolframscript); this is our own convenience.

Typed signatures

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

Domain: Statistics

Details

$PDF(x) = \dfrac{x^{k-1}e^{-x/\theta}}{\theta^k\,\Gamma(k)}$ , via Gamma. No domain clamp outside $x \ge 0$ .
$CDF(x) = 1 - Q(k, x/\theta)$ via the native two-argument GammaRegularized (not the three-argument generalized form @enumeratio/analytic adds — this has no dependency on that package), clamped to $0$ below $x=0$ .
$Mean = k\theta$ , $Variance = k\theta^2$ , both exact.
RandomVariate samples via Marsaglia–Tsang (shape $\ge 1$ ; boosted via a $Gamma(k+1)$ draw scaled by $U^{1/k}$ below $1$ ) — see RandomVariate for the seeded-PRNG divergence from Wolfram.