Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh (/ˈreɪli/).

Rayleigh

Probability density function
Cumulative distribution function
Parameters	scale: ${\displaystyle \sigma >0}$
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle {\frac {x}{\sigma ^{2}}}e^{-x^{2}/\left(2\sigma ^{2}\right)}}$
CDF	${\displaystyle 1-e^{-x^{2}/\left(2\sigma ^{2}\right)}}$
Quantile	${\displaystyle Q(F;\sigma )=\sigma {\sqrt {-2\ln(1-F)}}}$
Mean	${\displaystyle \sigma {\sqrt {\frac {\pi }{2}}}}$
Median	${\displaystyle \sigma {\sqrt {2\ln(2)}}}$
Mode	${\displaystyle \sigma }$
Variance	${\displaystyle {\frac {4-\pi }{2}}\sigma ^{2}}$
Skewness	${\displaystyle {\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}}$
Ex. kurtosis	${\displaystyle -{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}}$
Entropy	${\displaystyle 1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}}$
MGF	${\displaystyle 1+\sigma te^{\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left(\operatorname {erf} \left({\frac {\sigma t}{\sqrt {2}}}\right)+1\right)}$
CF	${\displaystyle 1-\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left(\operatorname {erfi} \left({\frac {\sigma t}{\sqrt {2}}}\right)-i\right)}$

A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (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 Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.