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 $erfi(z)$ 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: ${\frac {4-\pi }{2}}\sigma ^{2}$

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

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

Parameter estimation

The maximum likelihood estimate of the $\sigma$ parameter is given by:

$\sigma \approx {\sqrt {{\frac {1}{2N}}\sum _{i=0}^{N}x_{i}^{2}}}$

Related distributions

• $R\sim \mathrm {Rayleigh} (\sigma ^{2})$ 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.
• 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 $Y={\sqrt {2X\sigma \lambda }}\sim \mathrm {Rayleigh} (y|\sigma )$.

• If $R\sim \mathrm {Rayleigh} (\sigma ^{2})$ 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.

It would be nice to mention that, if the random variables X and Y are normal, independent with zero mean and equal variance (sigma^2), then the random variable Z = sqrt(X^2+Y^2) has a Rayleigh density with the parameter sigma. This gives some motivation to the use of the symbol "sigma" in the parameterization of the Rayleigh density.

See also

• Rayleigh fading