Rayleigh distribution

The probability density function of the Rayleigh distribution is[2]

${\displaystyle f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/(2\sigma ^{2})},\quad x\geq 0,}$

where ${\displaystyle \sigma }$ is the scale parameter of the distribution. The cumulative distribution function is[2]

${\displaystyle F(x;\sigma )=1-e^{-x^{2}/(2\sigma ^{2})}}$

for ${\displaystyle x\in [0,\infty ).}$

Consider the two-dimensional vector ${\displaystyle Y=(U,V)}$ which has components that are bivariate normally distributed, centered at zero, and independent. Then ${\displaystyle U}$ and ${\displaystyle V}$ have density functions

${\displaystyle f_{U}(x;\sigma )=f_{V}(x;\sigma )={\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sqrt {2\pi \sigma ^{2}}}}.}$

Let ${\displaystyle X}$ be the length of ${\displaystyle Y}$. That is, ${\displaystyle X={\sqrt {U^{2}+V^{2}}}.}$ Then ${\displaystyle X}$ has cumulative distribution function

${\displaystyle F_{X}(x;\sigma )=\iint _{D_{x}}f_{U}(u;\sigma )f_{V}(v;\sigma )\,dA,}$

where ${\displaystyle D_{x}}$ is the disk

${\displaystyle D_{x}=\left\{(u,v):{\sqrt {u^{2}+v^{2}}}\leq x\right\}.}$

Writing the double integral in polar coordinates, it becomes

${\displaystyle F_{X}(x;\sigma )={\frac {1}{2\pi \sigma ^{2}}}\int _{0}^{2\pi }\int _{0}^{x}re^{-r^{2}/(2\sigma ^{2})}\,dr\,d\theta ={\frac {1}{\sigma ^{2}}}\int _{0}^{x}re^{-r^{2}/(2\sigma ^{2})}\,dr.}$

Finally, the probability density function for ${\displaystyle X}$ is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is

${\displaystyle f_{X}(x;\sigma )={\frac {d}{dx}}F_{X}(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/(2\sigma ^{2})},}$

which is the Rayleigh distribution. It is straightforward to generalize to vectors of dimension other than 2. There are also generalizations when the components have unequal variance or correlations (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).[3]

The raw moments are given by:

${\displaystyle \mu _{j}=\sigma ^{j}2^{j/2}\,\Gamma \left(1+{\frac {j}{2}}\right),}$

where ${\displaystyle \Gamma (z)}$ is the gamma function.

The mean of a Rayleigh random variable is thus :

${\displaystyle \mu (X)=\sigma {\sqrt {\frac {\pi }{2}}}\ \approx 1.253\ \sigma .}$

The standard deviation of a Rayleigh random variable is:

${\displaystyle \operatorname {std} (X)={\sqrt {\left(2-{\frac {\pi }{2}}\right)}}\sigma \approx 0.655\ \sigma }$

The variance of a Rayleigh random variable is :

${\displaystyle \operatorname {var} (X)=\mu _{2}-\mu _{1}^{2}=\left(2-{\frac {\pi }{2}}\right)\sigma ^{2}\approx 0.429\ \sigma ^{2}}$

The mode is ${\displaystyle \sigma ,}$ and the maximum pdf is

${\displaystyle f_{\max }=f(\sigma$ ;\sigma )={\frac {1}{\sigma }}e^{-1/2}\approx {\frac {0.606}{\sigma }}.}

The skewness is given by:

${\displaystyle \gamma _{1}={\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}\approx 0.631}$

The excess kurtosis is given by:

${\displaystyle \gamma _{2}=-{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}\approx 0.245}$

The characteristic function is given by:

${\displaystyle \varphi (t)=1-\sigma te^{-{\frac {1}{2}}\sigma ^{2}t^{2}}{\sqrt {\frac {\pi }{2}}}\left[\operatorname {erfi} \left({\frac {\sigma t}{\sqrt {2}}}\right)-i\right]}$

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

${\displaystyle M(t)=1+\sigma t\,e^{{\frac {1}{2}}\sigma ^{2}t^{2}}{\sqrt {\frac {\pi }{2}}}\left[\operatorname {erf} \left({\frac {\sigma t}{\sqrt {2}}}\right)+1\right]}$

where ${\displaystyle \operatorname {erf} (z)}$ is the error function.

Given a sample of N independent and identically distributed Rayleigh random variables ${\displaystyle x_{i}}$ with parameter ${\displaystyle \sigma }$,

${\displaystyle {\widehat {\sigma }}^{2}=\!\,{\frac {1}{2N}}\sum _{i=1}^{N}x_{i}^{2}}$ is the maximum likelihood estimate and also is unbiased.

${\displaystyle {\widehat {\sigma }}\approx {\sqrt {{\frac {1}{2N}}\sum _{i=1}^{N}x_{i}^{2}}}}$ is a biased estimator that can be corrected via the formula

${\displaystyle \sigma ={\widehat {\sigma }}{\frac {\Gamma (N){\sqrt {N}}}{\Gamma (N+{\frac {1}{2}})}}={\widehat {\sigma }}{\frac {4^{N}N!(N-1)!{\sqrt {N}}}{(2N)!{\sqrt {\pi }}}}}$[4]

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

${\displaystyle X=\sigma {\sqrt {-2\ln U}}\,}$

has a Rayleigh distribution with parameter ${\displaystyle \sigma }$. This is obtained by applying the inverse transform sampling-method.

• ${\displaystyle R\sim \mathrm {Rayleigh} (\sigma )}$ is Rayleigh distributed if ${\displaystyle R={\sqrt {X^{2}+Y^{2}}}}$, where ${\displaystyle X\sim N(0,\sigma ^{2})}$ and ${\displaystyle Y\sim N(0,\sigma ^{2})}$ are independent normal random variables.[6] This gives motivation to the use of the symbol ${\displaystyle \sigma }$ in the above parametrization of the Rayleigh density.

• The magnitude ${\displaystyle |z|}$ of a standard complex normally distributed variable z is Rayleigh distributed.

• The chi distribution with v = 2 is equivalent to the Rayleigh Distribution with σ = 1.

• If ${\displaystyle R\sim \mathrm {Rayleigh} (1)}$, then ${\displaystyle R^{2}}$ has a chi-squared distribution with parameter ${\displaystyle N}$, degrees of freedom, equal to two (N = 2)

${\displaystyle [Q=R^{2}]\sim \chi ^{2}(N)\ .}$

• If ${\displaystyle R\sim \mathrm {Rayleigh} (\sigma )}$, then ${\displaystyle \sum _{i=1}^{N}R_{i}^{2}}$ has a gamma distribution with parameters ${\displaystyle N}$ and ${\displaystyle {\frac {1}{2\sigma ^{2}}}}$

${\displaystyle \left[Y=\sum _{i=1}^{N}R_{i}^{2}\right]\sim \Gamma (N,{\frac {1}{2\sigma ^{2}}}).}$

• The Rice distribution is a noncentral generalization of the Rayleigh distribution: ${\displaystyle \mathrm {Rayleigh} (\sigma )=\mathrm {Rice} (0,\sigma )}$.
• The Weibull distribution with the shape parameter k=2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter ${\displaystyle \sigma }$ is related to the Weibull scale parameter according to ${\displaystyle \lambda =\sigma {\sqrt {2}}.}$
• The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
• If ${\displaystyle X}$ has an exponential distribution ${\displaystyle X\sim \mathrm {Exponential} (\lambda )}$, then ${\displaystyle Y={\sqrt {X}}\sim \mathrm {Rayleigh} (1/{\sqrt {2\lambda }}).}$

• The half-normal distribution is the univariate special case of the Rayleigh distribution.

An application of the estimation of σ can be found in magnetic resonance imaging (MRI). As MRI images are recorded as complex images but most often viewed as magnitude images, the background data is Rayleigh distributed. Hence, the above formula can be used to estimate the noise variance in an MRI image from background data.[7] [8]

The Rayleigh distribution was also employed in the field of nutrition for linking dietary nutrient levels and human and animal responses. In this way, the parameter σ may be used to calculate nutrient response relationship.[9]

In the field of ballistics, the Rayleigh distribution is used for calculating the circular error probable - a measure of a weapon's precision.

In physical oceanography, the distribution of significant wave height approximately follows a Rayleigh distribution.[10]

• Circular error probable
• Rayleigh fading
• Rayleigh mixture distribution
• Rice distribution