RayleighDistribution

The Rayleigh distribution with scale $\sigma$ — the distribution of the magnitude of a 2D Gaussian vector.

RayleighDistribution(sigma)an inert distribution object — carries $\sigma$, unevaluated.

Typed signatures

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

Domain: Statistics

Details

• $PDF(x) = (x/\sigma^2)e^{-x^2/(2\sigma^2)}$ for $x \geq 0$.
• $CDF(x) = 1-e^{-x^2/(2\sigma^2)}$, clamped to $0$ below $x=0$.
• $Mean = \sigma\sqrt{\pi/2}$, $Variance = (4-\pi)\sigma^2/2$, both exact.
• RandomVariate samples via inverse transform: $\sigma\sqrt{-2\log U}$.

Examples

Basic #

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In$\operatorname{PDF}(\mathrm{RayleighDistribution}(3), 3)$Out$\frac{1}{3\sqrt{e}}$

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In$\operatorname{CDF}(\mathrm{RayleighDistribution}(3), 3)$Out$1-\sqrt{\frac{1}{e}}$

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In$\mathrm{mean}(\mathrm{RayleighDistribution}(3))$Out$\frac{1}{2}(3\sqrt{2}\sqrt{π})$

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In$\mathrm{Variance}(\mathrm{RayleighDistribution}(3))$Out$\frac{9(4-π)}{2}$

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In$\mathrm{median}(\mathrm{RayleighDistribution}(3))$Out$3\sqrt{2}\sqrt{\ln(2)}$

About

Examples	5
Checked against	Wolfram Language 1 agree

See also: ChiDistribution, MaxwellDistribution, PDF, CDF