Rayleigh 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 ( /ˈreɪlɪ/) is a continuous probability distribution. 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 speed is analyzed into its orthogonal 2-dimensional vector components. Assuming that the magnitude of each component is uncorrelated and normally distributed with equal variance, 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. In that case, the absolute value of the complex number is Rayleigh-distributed. The distribution is named after Lord Rayleigh.

The Rayleigh probability density function is

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

for parameter σ > 0, and cumulative distribution function

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

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

Properties

The raw moments are given by:

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

where Γ(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 σ and the maximum pdf is

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

The skewness is given by:

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

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:

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

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

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

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

Information entropy

The information entropy is given by

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

where γ is the Euler–Mascheroni constant.

Parameter estimation

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

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

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.

Generating Rayleigh-distributed random variates

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

$X=\sigma\sqrt{-2 \ln(1-U)}\,$

has a Rayleigh distribution with parameter σ. This follows from the form of the cumulative distribution function. Given that U is uniform, (1–U) has the same uniformity and the above may be simplified to

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

Note that if you are generating random numbers belonging to [0,1), exclude zero values to avoid the natural log of zero.

Related distributions

• R∼Rayleigh(σ) is Rayleigh distributed if $R = \sqrt{X^2 + Y^2}$, where X∼N(0,σ²) and Y∼N(0,σ²) 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∼Rayleigh(1), then R² has a chi-squared distribution with parameter N, degrees of freedom, equal to two (N=2) : $[Q=\sum_{i=1}^N R_i^2] \sim \chi^2(N)\$

• If R∼Rayleigh(σ), then $\sum_{i=1}^N R_i^2$ has a gamma distribution with parameters N and 2σ²: $[Y=\sum_{i=1}^N R_i^2] \sim \Gamma(N,2\sigma^2)$.

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

References

• Sijbers J., den Dekker A. J., J. Van Audekerke, Verhoye M. and Van Dyck D., "Estimation of the noise in magnitude MR images", Magnetic Resonance Imaging, Vol. 16, Nr. 1, p. 87–90, (1998)
• Sijbers J., den Dekker A. J., Raman E. and Van Dyck D., "Parameter estimation from magnitude MR images", International Journal of Imaging Systems and Technology, Vol. 10, Nr. 2, p. 109–114, (1999)

See also

• Rayleigh fading
• Rice distribution