Editing Rayleigh distribution (section)

==Related distributions==

* <math>R \sim \mathrm{Rayleigh}(\sigma)</math> is Rayleigh distributed if <math>R = \sqrt{X^2 + Y^2}</math>, where <math>X \sim N(0, \sigma^2)</math> and <math>Y \sim N(0, \sigma^2)</math> are independent [[Normal distribution|normal random variables]].<ref>[https://web.archive.org/web/20131105232146/http://home.kpn.nl/jhhogema1966/skeetn/ballist/sgs/sgs.htm#_Toc96439743 Hogema, Jeroen (2005) "Shot group statistics"]</ref> This gives motivation to the use of the symbol <math>\sigma</math> in the above parametrization of the Rayleigh density.
* The magnitude <math>|z|</math> of a [[standard complex normal distribution|standard complex normally distributed]] variable ''z'' is Rayleigh distributed.
* The [[chi distribution]] with ''v''&nbsp;=&nbsp;2 is equivalent to the Rayleigh Distribution with&nbsp;''&sigma;''&nbsp;=&nbsp;1:  <math>R(\sigma) \sim \sigma\chi_2^{\,}\ .</math> 
* If <math>R \sim \mathrm{Rayleigh} (1)</math>, then <math>R^2</math> has a [[chi-squared distribution]] with 2 degrees of freedom:  <math>[Q=R(\sigma)^2] \sim \sigma^2\chi_2^2\ .</math>
* If <math>R \sim \mathrm{Rayleigh}(\sigma)</math>, then <math>\sum_{i=1}^N R_i^2</math> has a [[gamma distribution]] with integer scale parameter <math>N</math> and rate parameter <math>\frac{1}{2\sigma^2}</math>
*: <math>\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,\frac{1}{2\sigma^2}\right)</math> with integer shape parameter ''N'' and rate parameter <math>\frac{1}{2\sigma^2}.</math>
*: <math>\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,2\sigma^2\right)</math> with integer shape parameter ''N'' and scale parameter <math>2\sigma^2.</math>
* The [[Rice distribution]] is a [[noncentral distribution|noncentral generalization]] of the Rayleigh distribution: <math> \mathrm{Rayleigh}(\sigma) = \mathrm{Rice}(0,\sigma) </math>.
* The [[Weibull distribution]] with the [[shape parameter]] ''k''&nbsp;=&nbsp;2 yields a Rayleigh distribution.  Then the Rayleigh distribution parameter <math>\sigma</math> is related to the Weibull scale parameter according to <math>\lambda = \sigma \sqrt{2} .</math>
* If <math>X</math> has an [[exponential distribution]] <math>X \sim \mathrm{Exponential}(\lambda)</math>, then <math>Y=\sqrt{X} \sim \mathrm{Rayleigh}(1/\sqrt{2\lambda}) .</math>
* The [[half-normal distribution]] is the one-dimensional equivalent of the Rayleigh distribution.
* The [[Maxwell–Boltzmann distribution]] is the three-dimensional equivalent of the Rayleigh distribution.