The Rayleigh distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The Rayleigh variate with scale parameter b is equivalent to the Weibull variate with scale parameter b and shape parameter 2:  Rayleigh(b) ~ Weibull(b,2).

The Rayleigh variate with scale parameter 1 is equivalent to a ChiSquare variate with degrees of freedom 2:  Rayleigh(1) ~ ChiSquare(2).

Note that the Rayleigh command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Rayleigh}\left(b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{u{\&ExponentialE;}^{-\frac{u^2}{2b^2}}}{b^2}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{0.5{\&ExponentialE;}^{-\frac{0.1250000000}{b^2}}}{b^2}$

$\mathrm{Mean}\left(X\right)$

$\frac{b\sqrt{2}\sqrt{\mathrm{\pi }}}{2}$

$\mathrm{Variance}\left(X\right)$

$\left(2-\frac{\mathrm{\pi }}{2}\right)b^2$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.