Rayleigh distribution

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Rayleigh distribution

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(′rā·lē ′dis·trə′byü·shən)

(statistics) A normal distribution of two uncorrelated variates with the same variance.

Rayleigh distribution

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Rayleigh distribution

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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 $\sigma >0,$ and cumulative distribution function

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

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

Contents

1 Properties
- 1.1 Information entropy
2 Parameter estimation
3 Generating Rayleigh-distributed random variates
4 Related distributions
5 References
6 See also

Properties

The raw moments are given by:

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

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

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

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 $\gamma$ 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.^[1]^[2]

Generating Rayleigh-distributed random variates

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

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

has a Rayleigh distribution with parameter $\sigma$ .

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=\sum_{i=1}^N R_i^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 $Y=\sqrt{2X\sigma^2\lambda} \sim \mathrm{Rayleigh}(\sigma)$ .

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)

^ Sijbers 1998
^ Sijbers 1999

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Bose–Einstein
Burr
chi-squared
chi
Coxian
Dagum
Davis
Erlang
exponential
F
Fermi–Dirac
folded normal
Fréchet
Gamma
generalized inverse Gaussian
half-logistic
half-normal
Hotelling's T-squared
hyper-exponential
hypoexponential
inverse chi-squared (scaled-inverse-chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell speed
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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