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Statistics[Distributions]

 Rayleigh
 Rayleigh distribution

 Calling Sequence Rayleigh(b) RayleighDistribution(b)

Parameters

 b - scale parameter

Description

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

$f\left(t\right)=\mathrm{piecewise}\left(t<0,0,\frac{t{ⅇ}^{-\frac{1}{2}\frac{{t}^{2}}{{b}^{2}}}}{{b}^{2}}\right)$

 subject to the following conditions:

$0

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Rayleigh}\left(b\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{u}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}\frac{{{u}}^{{2}}}{{{b}}^{{2}}}}}{{{b}}^{{2}}}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.5}{}{{ⅇ}}^{{-}\frac{{0.1250000000}}{{{b}}^{{2}}}}}{{{b}}^{{2}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{1}}{{2}}{}{b}{}\sqrt{{2}}{}\sqrt{{\mathrm{π}}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\left({2}{-}\frac{{1}}{{2}}{}{\mathrm{π}}\right){}{{b}}^{{2}}$ (4)

References

 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.