Editing Rayleigh distribution (section)

{{Distinguish|Rayleigh mixture distribution}}
{{short description|Probability distribution}}
{{Probability distribution|
  name       =Rayleigh|
  type       =density|
  pdf_image =[[Image:Rayleigh distributionPDF.svg|325px|Plot of the Rayleigh PDF]]<br /><small></small>|
  cdf_image  =[[Image:Rayleigh distributionCDF.svg|325px|Plot of the Rayleigh CDF]]<br /><small></small>|
  parameters =scale: <math>\sigma>0</math>|
  support    =<math>x\in [0,\infty)</math>|
  pdf        =<math>\frac{x}{\sigma^2} e^{-x^2/\left(2\sigma^2\right)}</math>|
  cdf        =<math>1 - e^{-x^2/\left(2\sigma^2\right)}</math>|
  quantile   =<math>Q(F;\sigma)=\sigma \sqrt{-2\ln(1 - F)}</math>|
  mean       =<math>\sigma \sqrt{\frac{\pi}{2}}</math>|
  median     =<math>\sigma\sqrt{2\ln(2)}</math>|
  mode       =<math>\sigma</math>|
  variance   =<math>\frac{4 - \pi}{2} \sigma^2</math>|
  skewness   =<math>\frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}</math>|
  kurtosis   =<math>-\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}</math>|
  entropy    =<math>1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}</math>|
  mgf        =<math>1+\sigma te^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right)</math>|
  char       =<math>1 - \sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\operatorname{erfi} \left(\frac{\sigma t}{\sqrt{2}}\right) - i\right)</math>|
}}

In [[probability theory]] and [[statistics]], the '''Rayleigh distribution''' is a [[continuous probability distribution]] for nonnegative-valued [[random variable]]s. Up to rescaling, it coincides with the [[chi distribution]] with two [[degrees of freedom]].
The distribution is named after [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] ({{IPAc-en|ˈ|r|eɪ|l|i}}).<ref>"The Wave Theory of Light", ''Encyclopedic Britannica'' 1888; "The Problem of the Random Walk", ''Nature'' 1905 vol.72 p.318</ref>

A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional [[Euclidean vector#Decomposition|components]]. One example where the Rayleigh distribution naturally arises is when [[wind]] velocity is analyzed in [[Plane (geometry)|two dimensions]].
Assuming that each component is [[uncorrelated]], [[Normal distribution|normally distributed]] with equal [[variance]], and zero [[mean]], which is infrequent, then the overall wind speed ([[Euclidean vector|vector]] magnitude) will be characterized by a Rayleigh distribution.   
A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed [[normal distribution|Gaussian]] with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.