View source for Rayleigh distribution

{{short description|Probability distribution}}
{{Distinguish|Rayleigh mixture distribution}}
{{Probability distribution|
  name       =Rayleigh|
  type       =density|
  pdf_image =[[Image:Rayleigh distributionPDF.svg|325px|Plot of the Rayleigh PDF]]<br />|
  cdf_image  =[[Image:Rayleigh distributionCDF.svg|325px|Plot of the Rayleigh CDF]]<br />|
  parameters =scale: <math>\sigma>0</math>|
  support    =<math>x\in [0,\infty)</math>|
  pdf        =<math>\frac{x}{\sigma^2} e^{-x^2/\left(2\sigma^2\right)}</math>|
  cdf        =<math>1 - e^{-x^2/\left(2\sigma^2\right)}</math>|
  quantile   =<math>Q(F;\sigma)=\sigma \sqrt{-2\ln(1 - F)}</math>|
  mean       =<math>\sigma \sqrt{\frac{\pi}{2}}</math>|
  median     =<math>\sigma\sqrt{2\ln(2)}</math>|
  mode       =<math>\sigma</math>|
  variance   =<math>\frac{4 - \pi}{2} \sigma^2</math>|
  skewness   =<math>\frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}</math>|
  kurtosis   =<math>-\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}</math>|
  entropy    =<math>1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}</math>|
  mgf        =<math>1+\sigma te^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right)</math>|
  char       =<math>1 - \sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\operatorname{erfi} \left(\frac{\sigma t}{\sqrt{2}}\right) - i\right)</math>|
}}

In [[probability theory]] and [[statistics]], the '''Rayleigh distribution''' is a [[continuous probability distribution]] for nonnegative-valued [[random variable]]s. Up to rescaling, it coincides with the [[chi distribution]] with two [[degrees of freedom]].
The distribution is named after [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] ({{IPAc-en|ˈ|r|eɪ|l|i}}).<ref>"The Wave Theory of Light", ''Encyclopedic Britannica'' 1888; "The Problem of the Random Walk", ''Nature'' 1905 vol.72 p.318</ref>

A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional [[Euclidean vector#Decomposition|components]]. One example where the Rayleigh distribution naturally arises is when [[wind]] velocity is analyzed in [[Plane (geometry)|two dimensions]].
Assuming that each component is [[uncorrelated]], [[Normal distribution|normally distributed]] with equal [[variance]], and zero [[mean]], which is infrequent, then the overall wind speed ([[Euclidean vector|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 [[normal distribution|Gaussian]] with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.

==Definition==

The [[probability density function]] of the Rayleigh distribution is<ref Name=PP>Papoulis, Athanasios; Pillai, S. (2001) ''Probability, Random Variables and Stochastic Processes p. 169''. {{isbn|0073660116}}, {{isbn|9780073660110}}</ref>

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

where <math>\sigma</math> is the [[scale parameter]] of the distribution. The [[cumulative distribution function]] is<ref Name=PP/>

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

for <math>x \in [0,\infty).</math>

==Relation to random vector length==

Consider the two-dimensional vector <math> Y = (U,V) </math> which has components that are [[bivariate normal distribution|bivariate normally distributed]], centered at zero, with equal variances <math>\sigma^2</math>, and independent. Then <math>U</math> and <math>V</math> have density functions

:<math>f_U(x; \sigma) = f_V(x;\sigma) = \frac{e^{-x^2/(2\sigma^2)}}{\sqrt{2\pi\sigma^2}}.</math>

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

:<math>F_X(x; \sigma) = \iint_{D_x} f_U(u;\sigma) f_V(v;\sigma) \,dA,</math>

where <math>D_x</math> is the disk

:<math>D_x = \left\{(u,v) : \sqrt{u^2 + v^2} \leq x\right\}.</math>

Writing the [[multiple integral|double integral]] in [[polar coordinate system|polar coordinates]], it becomes

:<math>F_X(x; \sigma) = \frac{1}{2\pi\sigma^2} \int_0^{2\pi} \int_0^x r e^{-r^2/(2\sigma^2)} \,dr\,d\theta = \frac 1 {\sigma^2} \int_0^x r e^{-r^2/(2\sigma^2)} \,dr.
</math>

Finally, the probability density function for <math>X</math> is the derivative of its cumulative distribution function, which by the [[fundamental theorem of calculus]] is

:<math>f_X(x;\sigma) = \frac d {dx} F_X(x;\sigma) = \frac x {\sigma^2} e^{-x^2/(2\sigma^2)},</math>

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 ([[Hoyt distribution]]), or when the vector ''Y'' follows a [[multivariate t-distribution|bivariate Student ''t''-distribution]] (see also: [[Hotelling's T-squared distribution]]).<ref>{{cite journal|last=Röver|first=C.|title=Student-t based filter for robust signal detection|journal=Physical Review D|volume=84|issue=12|year=2011|article-number=122004|doi=10.1103/physrevd.84.122004|arxiv=1109.0442|bibcode=2011PhRvD..84l2004R}}</ref>

{{Collapse top|title=Generalization to bivariate Student's t-distribution}}
{{anchor|Student's}}
Suppose <math>Y</math> is a random vector with components <math>u,v</math> that follows a [[multivariate t-distribution]]. If the components both have mean zero, equal variance and are independent, the bivariate Student's-t distribution takes the form:

:<math>f(u,v) = {1\over{2\pi\sigma^{2}}}\left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}</math>

Let <math>R = \sqrt{U^{2}+V^{2}}</math> be the magnitude of <math>Y</math>. Then the [[cumulative distribution function]] (CDF) of the magnitude is:

:<math> F(r) = {1\over{2\pi\sigma^{2}}}\iint_{D_{r}} \left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}du \; dv </math>

where <math>D_{r}</math> is the disk defined by:

:<math> D_{r} = \left\{ (u,v) : \sqrt{u^{2}+v^{2}} \leq r \right\} </math>

Converting to [[polar coordinates]] leads to the CDF becoming:

:<math> \begin{aligned} F(r) &= {1\over{2\pi\sigma^{2}}}\int_{0}^{r}\int_{0}^{2\pi} \rho\left( 1 + {\rho^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}d\theta \; d\rho \\ &= {1\over{\sigma^{2}}}\int_{0}^{r}\rho\left( 1 + {\rho^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} d\rho \\ &= 1-\left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2}  \end{aligned} </math>

Finally, the [[probability density function]] (PDF) of the magnitude may be derived:

:<math> f(r) = F'(r) = {r\over{\sigma^{2}}} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} </math>

In the limit as <math> \nu \rightarrow \infty </math>, the Rayleigh distribution is recovered because:

:<math> \lim_{\nu\rightarrow \infty} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} = e^{-r^{2}/2\sigma^{2}} </math>

{{Collapse bottom}}

==Properties==
The [[moment (mathematics)|raw moments]] are given by:
: <math>\mu_j = \sigma^j2^{j/2}\,\Gamma\left(1 + \frac j 2\right),</math> 
where <math>\Gamma(z)</math> is the [[gamma function]].

The [[mean]] of a  Rayleigh  random variable  is thus <!--(<math>k=1, \Gamma\left(\tfrac32\right) = \tfrac12 \sqrt{\pi}\,</math>)-->:
:<math>\mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253\ \sigma.</math>

The [[standard deviation]] of a  Rayleigh  random variable  is:

:<math>\operatorname{std}(X) = \sqrt{\left (2-\frac{\pi}{2}\right)} \sigma \approx 0.655\ \sigma</math>

The [[variance]]   of a  Rayleigh  random variable  is :

:<math>\operatorname{var}(X) = \mu_2-\mu_1^2 =  \left(2-\frac{\pi}{2}\right) \sigma^2 \approx 0.429\ \sigma^2</math>

The [[mode (statistics)|mode]] is <math>\sigma,</math> and the maximum pdf is

:<math> f_{\max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-1/2} \approx \frac{0.606}{\sigma}.</math>

The [[skewness]] is given by:

:<math>\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^{3/2}} \approx 0.631</math>

The excess [[kurtosis]] is given by:

:<math>\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245</math>

The [[characteristic function (probability theory)|characteristic function]] is given by:

:<math>\varphi(t) = 1 - \sigma te^{-\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\operatorname{erfi}\left(\frac{\sigma t}{\sqrt{2}}\right) - i\right]</math>

where <math>\operatorname{erfi}(z)</math> is the imaginary [[error function]]. The [[moment generating function]] is given by

:<math>
  M(t) = 1 + \sigma t\,e^{\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}}
           \left[\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right]</math>

where <math>\operatorname{erf}(z)</math> is the [[error function]].

===Differential entropy===

The [[differential entropy]] is given by{{Citation needed|date=April 2013}}

:<math>H = 1 + \ln\left(\frac \sigma {\sqrt{2}}\right) + \frac \gamma 2 </math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]].

== Parameter estimation ==

Given a sample of ''N'' [[independent and identically distributed]] Rayleigh random variables <math>x_i</math> with parameter <math>\sigma</math>,

: <math>\widehat{\sigma^2} = \!\,\frac{1}{2N}\sum_{i=1}^N x_i^2</math> is the [[maximum likelihood estimation|maximum likelihood]] estimate and also is [[bias of an estimator|unbiased]].

:<math>\widehat{\sigma}\approx \sqrt{\frac 1 {2N} \sum_{i=1}^N x_i^2}</math> is a biased estimator that can be corrected via the formula

:<math>\sigma = \widehat{\sigma} \frac {\Gamma(N)\sqrt{N}} {\Gamma\left(N + \frac 1 2\right)} = \widehat{\sigma} \frac {4^N N!(N-1)!\sqrt{N}} {(2N)!\sqrt{\pi}}</math><ref>[https://archive.org/details/jresv68Dn9p1005 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]</ref> <math>= \frac{\hat{\sigma}}{c_4(2N+1)}</math>, where [[Standard deviation#Unbiased sample standard deviation|c<sub>4</sub> is the correction factor used to unbias estimates of standard deviation for normal random variables]].

=== Confidence intervals ===
To find the (1&nbsp;&minus;&nbsp;''α'') confidence interval, first find the bounds <math>[a,b]</math> where:
:&nbsp; <math>P\left(\chi_{2N}^2 \leq a\right) = \alpha/2, \quad P\left(\chi_{2N}^2 \leq b\right) = 1 - \alpha/2</math>
then the scale parameter will fall within the bounds
:&nbsp; <math>\frac{{N}\overline{x^2}}{b} \leq {\widehat{\sigma^2}} \leq \frac{{N}\overline{x^2}}{a}</math><ref>[http://nvlpubs.nist.gov/nistpubs/jres/66D/jresv66Dn2p167_A1b.pdf 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]</ref>

== Generating random variates ==

Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] in the interval <nowiki>(0,&nbsp;1)</nowiki>, then the variate

:<math>X=\sigma\sqrt{-2 \ln U}\,</math>

has a Rayleigh distribution with parameter <math>\sigma</math>. This is obtained by applying the [[inverse transform sampling]]-method.

==Related distributions==

* <math>R \sim \mathrm{Rayleigh}(\sigma)</math> is Rayleigh distributed if <math>R = \sqrt{X^2 + Y^2}</math>, where <math>X \sim N(0, \sigma^2)</math> and <math>Y \sim N(0, \sigma^2)</math> are independent [[Normal distribution|normal random variables]].<ref>[https://web.archive.org/web/20131105232146/http://home.kpn.nl/jhhogema1966/skeetn/ballist/sgs/sgs.htm#_Toc96439743 Hogema, Jeroen (2005) "Shot group statistics"]</ref> This gives motivation to the use of the symbol <math>\sigma</math> in the above parametrization of the Rayleigh density.
* The magnitude <math>|z|</math> of a [[standard complex normal distribution|standard complex normally distributed]] variable ''z'' is Rayleigh distributed.
* The [[chi distribution]] with ''v''&nbsp;=&nbsp;2 is equivalent to the Rayleigh Distribution with&nbsp;''&sigma;''&nbsp;=&nbsp;1:  <math>R(\sigma) \sim \sigma\chi_2^{\,}\ .</math> 
* If <math>R \sim \mathrm{Rayleigh} (1)</math>, then <math>R^2</math> has a [[chi-squared distribution]] with 2 degrees of freedom:  <math>[Q=R(\sigma)^2] \sim \sigma^2\chi_2^2\ .</math>
* If <math>R \sim \mathrm{Rayleigh}(\sigma)</math>, then <math>\sum_{i=1}^N R_i^2</math> has a [[gamma distribution]] with integer scale parameter <math>N</math> and rate parameter <math>\frac{1}{2\sigma^2}</math>
*: <math>\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,\frac{1}{2\sigma^2}\right)</math> with integer shape parameter ''N'' and rate parameter <math>\frac{1}{2\sigma^2}.</math>
*: <math>\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,2\sigma^2\right)</math> with integer shape parameter ''N'' and scale parameter <math>2\sigma^2.</math>
* The [[Rice distribution]] is a [[noncentral distribution|noncentral generalization]] of the Rayleigh distribution: <math> \mathrm{Rayleigh}(\sigma) = \mathrm{Rice}(0,\sigma) </math>.
* The [[Weibull distribution]] with the [[shape parameter]] ''k''&nbsp;=&nbsp;2 yields a Rayleigh distribution.  Then the Rayleigh distribution parameter <math>\sigma</math> is related to the Weibull scale parameter according to <math>\lambda = \sigma \sqrt{2} .</math>
* If <math>X</math> has an [[exponential distribution]] <math>X \sim \mathrm{Exponential}(\lambda)</math>, then <math>Y=\sqrt{X} \sim \mathrm{Rayleigh}(1/\sqrt{2\lambda}) .</math>
* The [[half-normal distribution]] is the one-dimensional equivalent of the Rayleigh distribution.
* The [[Maxwell–Boltzmann distribution]] is the three-dimensional equivalent of the Rayleigh distribution.

== Applications ==

An application of the estimation of σ can be found in [[magnetic resonance imaging]] (MRI). As MRI images are recorded as [[complex numbers|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.<ref>{{cite journal | last1 = Sijbers | first1 = J. | last2 = den Dekker | first2 = A. J. | last3 = Raman | first3 = E. | last4 = Van Dyck | first4 = D. | year = 1999 | title = Parameter estimation from magnitude MR images | journal = International Journal of Imaging Systems and Technology | volume = 10 | issue = 2| pages = 109–114 | doi=10.1002/(sici)1098-1098(1999)10:2<109::aid-ima2>3.0.co;2-r| citeseerx = 10.1.1.18.1228 }}</ref>
<ref>{{cite journal | last1 = den Dekker | first1 = A. J. | last2 = Sijbers | first2 = J. | year = 2014 | title = Data distributions in magnetic resonance images: a review | journal = Physica Medica | volume = 30 | issue = 7| pages = 725–741 | doi=10.1016/j.ejmp.2014.05.002| pmid = 25059432 }}</ref>

The Rayleigh distribution was also employed in the field of [[nutrition]] for linking [[Diet (nutrition)|dietary]] [[nutrient]] levels and [[human]] and [[Animal husbandry|animal]] responses. In this way, the [[parameter]] σ may be used to calculate nutrient response relationship.<ref>{{Cite journal|last=Ahmadi|first=Hamed|date=2017-11-21|title=A mathematical function for the description of nutrient-response curve|journal=PLOS ONE|volume=12|issue=11|article-number=e0187292|doi=10.1371/journal.pone.0187292|pmid=29161271|issn=1932-6203|bibcode=2017PLoSO..1287292A|pmc=5697816|doi-access=free}}</ref>

In the field of [[ballistics]], the Rayleigh distribution is used for calculating the [[circular error probable]]—a measure of a gun's precision.

In [[physical oceanography]], the distribution of [[significant wave height]] approximately follows a Rayleigh distribution.<ref>{{Cite book |last=Dean |first=Robert G. |title=Water Wave Mechanics for Engineers and Scientists |last2=Dalrymple |first2=Robert A. |publisher=World Scientific |year=1991 |isbn=9-789810-204211 |series=Advanced Series on Ocean Engineering - Volume 2}}</ref><ref>{{Cite web|title=Rayleigh Probability Distribution Applied to Random Wave Heights|url=https://www.usna.edu/NAOE/_files/documents/Courses/EN330/Rayleigh-Probability-Distribution-Applied-to-Random-Wave-Heights.pdf|publisher=United States Naval Academy}}</ref>

==See also==
*[[Circular error probable]]
*[[Rayleigh fading]]
*[[Rayleigh mixture distribution]]
*[[Rice distribution]]

== References ==

{{reflist}}

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Rayleigh Distribution}}
[[Category:Continuous distributions]]
[[Category:Exponential family distributions]]