Rayleigh distribution

Rayleigh
	Probability density function;
	Cumulative distribution function;
Parameters	scale:
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for positive-valued random variables. It is a chi distribution in two degrees of freedom.

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 in two dimensions. Assuming that each component is 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 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 (/ˈreɪli/).^[1]

Definition

The probability density function of the Rayleigh distribution is^[2]

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

where $\sigma$ is the scale parameter of the distribution. The cumulative distribution function is^[2]

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

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

Relation to random vector length

Consider the two-dimensional vector $Y=(U,V)$ which has components that are normally distributed, centered at zero, and independent. Then $U$ and $V$ have density functions

f_{U}(x;\sigma )=f_{V}(x;\sigma )={\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sqrt {2\pi \sigma ^{2}}}}.

Let $X$ be the length of $Y$ . That is, $X={\sqrt {U^{2}+V^{2}}}.$ Then $X$ has cumulative distribution function

F_{X}(x;\sigma )=\iint _{D_{x}}f_{U}(u;\sigma )f_{V}(v;\sigma )\,dA,

where $D_{x}$ is the disk

D_{x}=\left\{(u,v):{\sqrt {u^{2}+v^{2}}}<x\right\}.

Writing the double integral in polar coordinates, it becomes

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 $X$ is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is

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, or when the vector Y follows a bivariate Student t-distribution.^[3]

Properties

The raw moments are given by:

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

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

The mean of a Rayleigh random variable is thus :

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

The variance of a Rayleigh random variable is :

\operatorname {var} (X)=\mu _{2}-\mu _{1}^{2}=\left(2-{\frac {\pi }{2}}\right)\sigma ^{2}\approx 0.429\ \sigma ^{2}

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

f_{\max }=f(\sigma ;\sigma )={\frac {1}{\sigma }}e^{-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^{-{\frac {1}{2}}\sigma ^{2}t^{2}}{\sqrt {\frac {\pi }{2}}}\left[\operatorname {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[\operatorname {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

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 the maximum likelihood estimate and also is unbiased.

{\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

\sigma ={\widehat {\sigma }}{\frac {\Gamma (N){\sqrt {N}}}{\Gamma (N+{\frac {1}{2}})}}={\widehat {\sigma }}{\frac {4^{N}N!(N-1)!{\sqrt {N}}}{(2N)!{\sqrt {\pi }}}}

^[4]

Confidence intervals

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

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

then the scale parameter will fall within the bounds

{\frac {{2N}{\overline {x^{2}}}}{b}}\leq {\sigma }^{2}\leq {\frac {{2N}{\overline {x^{2}}}}{a}}

^[5]

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.^[6] (This gives motivation to the use of the symbol "sigma" in the above parametrization of the Rayleigh density.)
The chi distribution with v = 2 is equivalent to the 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 $2\sigma ^{2}$

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

The Rice distribution is a generalization of the Rayleigh distribution: $\mathrm {Rayleigh} (\sigma )=\mathrm {Rice} (0,\sigma )$ .
The Weibull distribution with the "shape parameter" k=2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter $\sigma$ is related to the Weibull scale parameter according to $\lambda =\sigma {\sqrt {2}}.$
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 $Y={\sqrt {X}}\sim \mathrm {Rayleigh} (1/{\sqrt {2\lambda }}).$

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.^[7] ^[8] The principle of 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]

References

↑ "The Wave Theory of Light", Encyclopedic Britannica 1888; "The Problem of the Random Walk", Nature 1905 vol.72 p.318
1 2 Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. ISBN 0073660116, ISBN 9780073660110
↑ Röver, C. (2011). "Student-t based filter for robust signal detection". Physical Review D. 84 (12): 122004. arXiv:1109.0442. Bibcode:2011PhRvD..84l2004R. doi:10.1103/physrevd.84.122004.
↑ 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.; Van Dyck, D. (1999). "Parameter estimation from magnitude MR images". International Journal of Imaging Systems and Technology. 10 (2): 109–114. doi:10.1002/(sici)1098-1098(1999)10:2<109::aid-ima2>3.0.co;2-r.
↑ den Dekker, A. J.; Sijbers, J. (2014). "Data distributions in magnetic resonance images: a review". Physica Medica. 30 (7): 725–741. doi:10.1016/j.ejmp.2014.05.002.
↑ Ahmadi, Hamed (2017-11-21). "A mathematical function for the description of nutrient-response curve". PLOS ONE. 12 (11): e0187292. Bibcode:2017PLoSO..1287292A. doi:10.1371/journal.pone.0187292. ISSN 1932-6203.

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[1] "The Wave Theory of Light", Encyclopedic Britannica 1888; "The Problem of the Random Walk", Nature 1905 vol.72 p.318

[PP-2] 1 2 Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. ISBN 0073660116, ISBN 9780073660110

[3] Röver, C. (2011). "Student-t based filter for robust signal detection". Physical Review D. 84 (12): 122004. arXiv:1109.0442. Bibcode:2011PhRvD..84l2004R. doi:10.1103/physrevd.84.122004.

[4] 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

[5] 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

[6] Hogema, Jeroen (2005) "Shot group statistics"

[7] Sijbers, J.; den Dekker, A. J.; Raman, E.; Van Dyck, D. (1999). "Parameter estimation from magnitude MR images". International Journal of Imaging Systems and Technology. 10 (2): 109–114. doi:10.1002/(sici)1098-1098(1999)10:2<109::aid-ima2>3.0.co;2-r.

[8] Dekker, A. J.; Sijbers, J. (2014). "Data distributions in magnetic resonance images: a review". Physica Medica. 30 (7): 725–741. doi:10.1016/j.ejmp.2014.05.002.

[9] Ahmadi, Hamed (2017-11-21). "A mathematical function for the description of nutrient-response curve". PLOS ONE. 12 (11): e0187292. Bibcode:2017PLoSO..1287292A. doi:10.1371/journal.pone.0187292. ISSN 1932-6203.

Probability distributions
List
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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic 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 Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-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 asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Probability density function
Cumulative distribution function
Parameters	scale: $\sigma >0$
Support	$x\in [0,\infty )$
PDF	${\frac {x}{\sigma ^{2}}}e^{-x^{2}/\left(2\sigma ^{2}\right)}$
CDF	$1-e^{-x^{2}/\left(2\sigma ^{2}\right)}$
Quantile	$Q(F;\sigma )=\sigma {\sqrt {-2\ln(1-F)}}$
Mean	$\sigma {\sqrt {\frac {\pi }{2}}}$
Median	$\sigma {\sqrt {2\ln(2)}}$
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	$1+\sigma te^{\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left(\operatorname {erf} \left({\frac {\sigma t}{\sqrt {2}}}\right)+1\right)$
CF	$1-\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left(\operatorname {erfi} \left({\frac {\sigma t}{\sqrt {2}}}\right)-i\right)$