BetaDistribution

The Beta distribution on $[0, 1]$ with shape parameters $\alpha, \beta$ .

BetaDistribution(alpha, beta)an inert distribution object — carries its parameters, unevaluated. PDF, CDF, Mean, Variance and RandomVariate all read it.

Typed signatures

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

Domain: Statistics

Details

$PDF(x) = \dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$ , via Beta. No domain clamp outside $[0,1]$ — the formula is evaluated as written there too.
$CDF(x) = I_x(\alpha,\beta)$ via BetaRegularized, clamped to $0$ below $x=0$ and $1$ above $x=1$ .
$Mean = \dfrac{\alpha}{\alpha+\beta}$ , $Variance = \dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$ , both exact.
RandomVariate samples via two GammaDistribution draws (Marsaglia–Tsang), $X/(X+Y)$ — see RandomVariate for the seeded-PRNG divergence from Wolfram.