Contents

Rayleigh distribution

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	${\displaystyle {\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}}$
Excess 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	${\displaystyle 1-\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left(\operatorname {erfi} \left({\frac {\sigma t}{\sqrt {2}}}\right)-i\right)}$

Inprobability 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/).^[1]

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, which is infrequent, 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.

Definition[edit]

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 functionis^[2]

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

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

Relation to random vector length[edit]

Consider the two-dimensional vector ${\displaystyle Y=(U,V)}$ which has components that are bivariate normally distributed, centered at zero, with equal variances ${\displaystyle \sigma ^{2}}$, 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 integralinpolar 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 calculusis

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

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]

Properties[edit]

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 modeis${\displaystyle \sigma ,}$ 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.

Differential entropy[edit]

The differential entropy is given by^{[citation needed]}

${\displaystyle H=1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}}$

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.

Parameter estimation[edit]

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 \left(N+{\frac {1}{2}}\right)}}={\widehat {\sigma }}{\frac {4^{N}N!(N-1)!{\sqrt {N}}}{(2N)!{\sqrt {\pi }}}}} ^[4] ${\displaystyle ={\frac {\hat {\sigma }}{c_{4}(2N+1)}}}$, where c₄ is the correction factor used to unbias estimates of standard deviation for normal random variables.

Confidence intervals[edit]

To find the (1 − α) confidence interval, first find the bounds ${\displaystyle [a,b]}$ where:

  ${\displaystyle P\left(\chi _{2N}^{2}\leq a\right)=\alpha /2,\quad P\left(\chi _{2N}^{2}\leq b\right)=1-\alpha /2}$

then the scale parameter will fall within the bounds

  ${\displaystyle {\frac {{N}{\overline {x^{2}}}}{b}}\leq {\widehat {\sigma ^{2}}}\leq {\frac {{N}{\overline {x^{2}}}}{a}}}$^[5]

Generating random variates[edit]

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.

Related distributions[edit]

• ${\displaystyle R\sim \mathrm {Rayleigh} (\sigma )}$ 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: ${\displaystyle R(\sigma )\sim \sigma \chi _{2}^{\,}\ .}$
• If${\displaystyle R\sim \mathrm {Rayleigh} ($1)} , then ${\displaystyle R^{2}}$ has a chi-squared distribution with 2 degrees of freedom: ${\displaystyle [Q=R(\sigma )^{2}]\sim \sigma ^{2}\chi _{2}^{2}\ .}$
• 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 ${\displaystyle {\frac {1}{2\sigma ^{2}}}}$

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

• 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 ${\displaystyle \lambda =\sigma {\sqrt {2}}.}$
• 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 one-dimensional equivalent of the Rayleigh distribution.
• The Maxwell–Boltzmann distribution is the three-dimensional equivalent of the Rayleigh distribution.

Applications[edit]

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 gun's precision.

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

See also[edit]

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

References[edit]