Rayleigh distribution

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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	$1+\sigma t\,e^{{\sigma ^{2}t^{2}/2}}{\sqrt {{\frac {\pi }{2}}}}\left({\textrm {erf}}\left({\frac {\sigma t}{{\sqrt {2}}}}\right)\!+\!1\right)$
CF	$1\!-\!\sigma te^{{-\sigma ^{2}t^{2}/2}}{\sqrt {{\frac {\pi }{2}}}}\!\left({\textrm {erfi}}\!\left({\frac {\sigma t}{{\sqrt {2}}}}\right)\!-\!i\right)$

In probability theory and statistics, the Rayleigh distribution /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^[1]

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

where $\sigma >0,$ is the scale parameter of the distribution. The cumulative distribution function is^[1]

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

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

Relation to random vector lengths

Consider the two-dimensional vector $Y=(U,V)$ which has components that are Gaussian-distributed and independent. Then $f_{U}(u;\sigma )={\frac {e^{{-u^{2}/2\sigma ^{2}}}}{{\sqrt {2\pi \sigma ^{2}}}}}$ , and similarly for $f_{V}(v;\sigma )$ .

Let $x$ be the length of $Y$ . It is distributed as

$f(x;\sigma )={\frac {1}{2\pi \sigma ^{2}}}\int _{{-\infty }}^{\infty }du\,\int _{{-\infty }}^{\infty }dv\,e^{{-u^{2}/2\sigma ^{2}}}e^{{-v^{2}/2\sigma ^{2}}}\delta (x-{\sqrt {u^{2}+v^{2}}}).$

By transforming to the polar coordinate system one has

$f(x;\sigma )={\frac {1}{2\pi \sigma ^{2}}}\int _{0}^{{2\pi }}\,d\phi \int _{0}^{\infty }dr\,\delta (r-x)re^{{-r^{2}/2\sigma ^{2}}}={\frac {x}{\sigma ^{2}}}e^{{-x^{2}/2\sigma ^{2}}},$

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.

Properties

The raw moments are given by:

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

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:

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

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

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

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

Differential entropy

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

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

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

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

Confidence intervals

To find the (1 − α) confidence interval, first find $\chi _{1}^{2},\ \chi _{2}^{2}$ where:

$Pr(\chi ^{2}(2n)\leq \chi _{1}^{2})=\alpha /2,\quad Pr(\chi ^{2}(2n)\leq \chi _{2}^{2})=1-\alpha /2$

then

${\frac {2n\overline {x^{2}}}{\chi _{2}^{2}}}\leq \widehat {\sigma }^{2}\leq {\frac {2n\overline {x^{2}}}{\chi _{1}^{2}}}$ ^[3]

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.^[4] (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)

The chi distribution with v = 2 is equivalent to Rayleigh Distribution with σ = 1. I.e., 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 ${\frac {1}{2\sigma ^{2}}}$

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

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 $Y={\sqrt {2X\sigma ^{2}\lambda }}\sim {\mathrm {Rayleigh}}(\sigma ).$

Applications

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.^[5]

References

↑ 1.0 1.1 Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processe. ISBN 0073660116, ISBN 9780073660110
↑ Siddiqui, M. M. (1964) "Statistical inference for Rayleigh distributions", The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9, p. 1007
↑ Siddiqui, M. M. (1961) "Some Problems Connected With Rayleigh Distributions", The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2, p. 169
↑ Hogema, Jeroen (2005) "Shot group statistics"
↑ Sijbers J., den Dekker A. J., Raman E. and Van Dyck D. (1999) "Parameter estimation from magnitude MR images", International Journal of Imaging Systems and Technology, 10(2), 109–114

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
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–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
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 Johnson SU 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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