Rayleigh distribution

Not to be confused with Rayleigh mixture distribution.

Rayleigh

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

In probability theory and statistics, the Rayleigh distribution (pron.: /ˈreɪli/) is a continuous probability distribution for positive-valued random variables.

A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed into its orthogonal 2-dimensional vector components. Assuming that the magnitudes of each component are 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 i.i.d. (independently and identically distributed) Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.

The distribution is named after Lord Rayleigh.^{[citation needed]}

Definition

The probability density function of the Rayleigh distribution is

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

where $\sigma >0,$ is the scale parameter of the distribution. The cumulative distribution function is

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

for $x \in [0,\infty).$

Properties

The raw moments are given by:

$\mu_k = \sigma^k2^\frac{k}{2}\,\Gamma\left(1 + \frac{k}{2}\right)$

where $\Gamma(z)$ is the Gamma function.

The mean and variance of a Rayleigh random variable may be expressed as:

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

and

$\textrm{var}(X) = \frac{4 - \pi}{2} \sigma^2 \approx 0.429 \sigma^2$

The mode is $\sigma$ and the maximum pdf is

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

The skewness is given by:

$\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^\frac{3}{2}} \approx 0.631$

The excess kurtosis is given by:

$\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245$

The characteristic function is given by:

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

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

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

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

Information entropy

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

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

where $\gamma$ is the Euler–Mascheroni constant.

Parameter estimation

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

$\hat{\sigma}\approx \!\,\sqrt{\frac{1}{2N}\sum_{i=1}^N x_i^2}.$

An application of the estimation of $\sigma$ 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.

Generating random variates

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

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

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

Related distributions

• $R \sim \mathrm{Rayleigh}(\sigma)$ is Rayleigh distributed if $R = \sqrt{X^2 + Y^2}$, where $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ are independent normal random variables. (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-squared distribution with parameter $N$, degrees of freedom, equal to two (N=2) : $[Q=R^2] \sim \chi^2(N)\ .$

• 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 with v=2 is equivalent to Rayleigh Distribution with sigma=1
• The Rice distribution is a generalization of the Rayleigh distribution.

• The Weibull distribution is a generalization of the Rayleigh distribution. In this instance, parameter $\sigma$ is related to the Weibull scale parameter $\lambda$: $\lambda = \sigma \sqrt{2} .$
• The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
• If $X$ has an exponential distribution $X \sim \mathrm{Exponential}(\lambda)$, then $Y=\sqrt{2X\sigma^2\lambda} \sim \mathrm{Rayleigh}(\sigma) .$

See also

• Rayleigh fading
• Rayleigh mixture distribution