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Rayleigh Distribution

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RayleighDistribution

The distribution with probability density function and distribution function

P(r)=(re^(-r^2/(2s^2)))/(s^2)
(1)
D(r)=1-e^(-r^2/(2s^2))
(2)

for r in [0,infty) and parameter s.

It is implemented in the Wolfram Language as RayleighDistribution[s].

The raw moments are given by

mu_n^'=2^(n/2)s^nGamma(1+1/2n),
(3)

where Gamma(x) is the gamma function, giving the first few as

mu_0^'=1
(4)
mu_1^'=ssqrt(pi/2)
(5)
mu_2^'=2s^2
(6)
mu_3^'=3s^3sqrt(pi/2)
(7)
mu_4^'=8s^4.
(8)

The central moments are therefore

mu_2=(4-pi)/2s^2
(9)
mu_3=sqrt(pi/2)(pi-3)s^3
(10)
mu_4=(32-3pi^2)/4s^4.
(11)

The mean, variance, skewness, and kurtosis excess are

mu=ssqrt(pi/2)
(12)
sigma^2=(4-pi)/2s^2
(13)
gamma_1=(2(pi-3)sqrt(pi))/((4-pi)^(3/2))
(14)
gamma_2=-(6pi^2-24pi+16)/((pi-4)^2).
(15)

The characteristic function is

phi(t)=1-sqrt(pi/2)ste^(-s^2t^2/2)[erfi((st)/(sqrt(2)))-i].
(16)

See also

Maxwell Distribution

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References

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 104 and 148, 1984.

Referenced on Wolfram|Alpha

Rayleigh Distribution

Cite this as:

Weisstein, Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RayleighDistribution.html

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