ParetoDistribution

The Pareto distribution with minimum value $k$ and shape $\alpha$ .

ParetoDistribution(k, alpha)an inert distribution object — carries

k, \alpha

, unevaluated.

Typed signatures

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

Domain: Statistics

Details

$PDF(x) = \alpha k^\alpha / x^{\alpha+1}$ for $x \geq k$ .
$CDF(x) = 1-(k/x)^\alpha$ , clamped to $0$ below $x=k$ .
$Mean = \alpha k/(\alpha-1)$ — exact only for $\alpha>1$ ; a documented divergence for $\alpha \leq 1$ , where the mean is actually infinite.
$Variance = k^2\alpha/((\alpha-1)^2(\alpha-2))$ — exact only for $\alpha>2$ , same divergence policy.
RandomVariate samples via inverse transform: $k/U^{1/\alpha}$ .