Rayleigh distribution

From Wikipedia, the free encyclopedia

Rayleigh
Probability density function
Plot of the Rayleigh PDF
Cumulative distribution function
Plot of the Rayleigh CDF
Parameters \sigma>0\,
Support x\in [0;\infty)
Probability density function (pdf) \frac{x \exp\left(\frac{-x^2}{2\sigma^2}\right)}{\sigma^2}
Cumulative distribution function (cdf) 1-\exp\left(\frac{-x^2}{2\sigma^2}\right)
Mean \sigma \sqrt{\frac{\pi}{2}}
Median \sigma\sqrt{\ln(4)}\,
Mode \sigma\,
Variance \frac{4 - \pi}{2} \sigma^2
Skewness \frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}
Excess kurtosis -\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}
Entropy 1+\ln\left(\frac{1}{\sqrt{2}\sigma^3}\right)+\frac{\gamma}{2}
Moment-generating function (mgf) 1+\sigma t\,e^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right)
Characteristic function 1\!-\!\sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}\!\left(\textrm{erfi}\!\left(\frac{\sigma t}{\sqrt{2}}\right)\!-\!i\right)

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. It usually arises when a two-dimensional vector (e.g. wind velocity) has its two orthogonal components normally and independently distributed. The absolute value (e.g. wind speed) will then have a Rayleigh distribution. The distribution may also arise in the case of random complex numbers whose real and imaginary components are normally and independently distributed. The absolute value of these numbers will then be Rayleigh-distributed.

The probability density function is

f(x|\sigma) = \frac{x \exp\left(\frac{-x^2}{2\sigma^2}\right)}{\sigma^2}.

The characteristic function is given by:

\varphi(t)=
1\!-\!\sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}\!\left(\textrm{erfi}\!\left(\frac{\sigma t}{\sqrt{2}}\right)\!-\!i\right)

where \operatorname{erfi}(z) is the complex error function. The moment generating function is given by

M(t)=\,
1+\sigma t\,e^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right),

where erf(z) is the error function. The raw moments are then given by

\mu_k=\sigma^k2^{k/2}\,\Gamma(1+k/2)\,

where Γ(z) is the Gamma function. The moments may be used to calculate:

Mean: \sigma \sqrt{\frac{\pi}{2}}

Variance: \frac{4-\pi}{2} \sigma^2

Skewness: \frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}

Kurtosis: - \frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}

[edit] Parameter estimation

Given N independent and identically distributed Rayleigh random variables with parameter σ, the maximum likelihood estimate of σ is

\hat{\sigma}=\sqrt{\frac{1}{2N}\sum_{i=1}^N x_i^2}.

[edit] Related distributions

[edit] See also

Image:Bvn-small.png Probability distributionsview  talk  edit ]
Univariate Multivariate
Discrete: BenfordBernoullibinomialBoltzmanncategoricalcompound PoissondegenerateGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomialmultivariate Polya
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfadingFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHalf-LogisticHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-squareinverse Gaussianinverse gammaKumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speednormal (Gaussian)normal inverse GaussianParetoPearsonpolarraised cosineRayleighrelativistic Breit-WignerRiceshifted GompertzStudent's ttriangulartype-1 Gumbeltype-2 GumbeluniformVariance-GammaVoigtvon MisesWeibullWigner semicircleWilks' lambda Dirichletinverse-WishartKentmatrix normalmultivariate normalmultivariate Studentvon Mises-FisherWigner quasiWishart
Miscellaneous: Cantorconditionalexponential familyinfinitely divisiblelocation-scale familymarginalmaximum entropyphase-typeposteriorpriorquasisamplingsingular
In other languages