Rayleigh distribution

Rayleigh

Probability density function
Cumulative distribution function
Parameters	$\sigma >0\,$
Support	$x\in [0;\infty )$
PDF	${\frac {x\exp \left({\frac {-x^{2}}{2\sigma ^{2}}}\right)}{\sigma ^{2}}}$
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	$1+\ln \left({\frac {1}{{\sqrt {2}}\sigma ^{3}}}\right)+{\frac {\gamma }{2}}$
MGF	$1+\sigma t\,e^{\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left({\textrm {erf}}\left({\frac {\sigma t}{\sqrt {2}}}\right)\!+\!1\right)$
CF	$1\!-\!\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\!\left({\textrm {erfi}}\!\left({\frac {\sigma t}{\sqrt {2}}}\right)\!-\!i\right)$

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:

$\varphi (t)=$

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

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

$M(t)=\,$

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

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

$\mu _{k}=\sigma ^{k}2^{k/2}\,\Gamma (1+k/2)\,$

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

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

Variance: $2\sigma ^{2}$

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

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

Parameter estimation

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

${\hat {\sigma }}={\sqrt {{\frac {1}{2N}}\sum _{i=1}^{N}x_{i}^{2}}}.$

Related distributions

• $R\sim \mathrm {Rayleigh} (\sigma )$ is a Rayleigh distribution if $R={\sqrt {X^{2}+Y^{2}}}$ where $X\sim N(0,\sigma ^{2})$ and $Y\sim N(0,\sigma ^{2})$ are two independent normal distributions. (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
• If $R\sim \mathrm {Rayleigh} (1)$ then $R^{2}$ has a chi-square distribution with two degrees of freedom: $R^{2}\sim \chi _{2}^{2}$
• If $X$ has an exponential distribution $X\sim \mathrm {Exponential} (x|\lambda )$ then $Y={\sqrt {2X\sigma ^{2}\lambda }}\sim \mathrm {Rayleigh} (y|\sigma )$.

• If $R\sim \mathrm {Rayleigh} (\sigma )$ then $\sum _{i=1}^{N}R_{i}^{2}$ has a gamma distribution with parameters $N$ and $2\sigma ^{2}$: $[Y=\sum _{i=1}^{N}R_{i}^{2}]\sim \Gamma (N,2\sigma ^{2})$.

• The Chi distribution is a generalization of the Rayleigh distribution.
• The Rice distribution is a generalization of the Rayleigh distribution.
• The Weibull distribution is a generalization of the Rayleigh distribution.
• The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.

See also

• Rayleigh fading
• Rice distribution