WeibullDistribution

The Weibull distribution with shape $\alpha$ and scale $\beta$.

WeibullDistribution(alpha, beta)an inert distribution object — carries $\alpha, \beta$, unevaluated.

Typed signatures

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

Domain: Statistics

Details

• $PDF(x) = (\alpha/\beta)(x/\beta)^{\alpha-1}e^{-(x/\beta)^\alpha}$ for $x \geq 0$.
• $CDF(x) = 1-e^{-(x/\beta)^\alpha}$, clamped to $0$ below $x=0$.
• $Mean = \beta\Gamma(1+1/\alpha)$, $Variance = \beta^2(\Gamma(1+2/\alpha)-\Gamma(1+1/\alpha)^2)$, both exact.
• RandomVariate samples via inverse transform: $\beta(-\log(1-U))^{1/\alpha}$.

Examples

Basic #

Cross-checked against wolframscript's N[PDF[WeibullDistribution[2, 5], 4]]

live ↗

In$\operatorname{PDF}(\mathrm{WeibullDistribution}(2, 5), 4)$Out$\frac{8}{25\exp(\frac{16}{25})}$

Cross-checked against wolframscript's N[CDF[WeibullDistribution[2, 5], 4]]

live ↗

In$\operatorname{CDF}(\mathrm{WeibullDistribution}(2, 5), 4)$Out$1-\exp(-\frac{16}{25})$

live ↗

In$\mathrm{mean}(\mathrm{WeibullDistribution}(2, 5))$Out$5Γ(\frac{3}{2})$

The quantile at 1/2: $\beta \ln(2)^{1/\alpha}$

live ↗

In$\mathrm{median}(\mathrm{WeibullDistribution}(2, 5))$Out$5\sqrt{\ln(2)}$

Scope #

live ↗

In$\operatorname{CDF}(\mathrm{WeibullDistribution}(2, 5), N(4, 25))$Out$1-\exp(-\frac{16}{25})$

live ↗

In$\operatorname{PDF}(\mathrm{WeibullDistribution}(2, 5), N(4, 25))$Out$\frac{8}{25\exp(\frac{16}{25})}$

About

Examples	6
Checked against	Wolfram Language 3 agree

See also: ChiSquareDistribution, PDF, CDF