Rayleigh distribution#Student's t

The probability density function of the Rayleigh distribution isPapoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. {{isbn|0073660116}}, {{isbn|9780073660110}} {{Page needed|date=April 2013}}

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

where $\backslash sigma$ is the scale parameter of the distribution. The cumulative distribution function is

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

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

Consider the two-dimensional vector $Y\; =\; (U,V)$ which has components that are bivariate normally distributed, centered at zero, with equal variances $\backslash sigma^2$, and independent. Then $U$ and $V$ have density functions

:$f\_U(x;\; \backslash sigma)\; =\; f\_V(x;\backslash sigma)\; =\; \backslash frac\{e^\{-x^2/(2\backslash sigma^2)\}\}\{\backslash sqrt\{2\backslash pi\backslash sigma^2\}\}.$

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

:$F\_X(x;\; \backslash sigma)\; =\; \backslash iint\_\{D\_x\}\; f\_U(u;\backslash sigma)\; f\_V(v;\backslash sigma)\; \backslash ,dA,$

where $D\_x$ is the disk

:$D\_x\; =\; \backslash left\backslash \{(u,v)\; :\; \backslash sqrt\{u^2\; +\; v^2\}\; \backslash leq\; x\backslash right\backslash \}.$

Writing the double integral in polar coordinates, it becomes

:$F\_X(x;\; \backslash sigma)\; =\; \backslash frac\{1\}\{2\backslash pi\backslash sigma^2\}\; \backslash int\_0^\{2\backslash pi\}\; \backslash int\_0^x\; r\; e^\{-r^2/(2\backslash sigma^2)\}\; \backslash ,dr\backslash ,d\backslash theta\; =\; \backslash frac\; 1\; \{\backslash sigma^2\}\; \backslash int\_0^x\; r\; e^\{-r^2/(2\backslash sigma^2)\}\; \backslash ,dr.$

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

:$f\_X(x;\backslash sigma)\; =\; \backslash frac\; d\; \{dx\}\; F\_X(x;\backslash sigma)\; =\; \backslash frac\; x\; \{\backslash sigma^2\}\; e^\{-x^2/(2\backslash sigma^2)\},$

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 (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).{{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|pages=122004|doi=10.1103/physrevd.84.122004|arxiv=1109.0442|bibcode=2011PhRvD..84l2004R}}

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Suppose $Y$ is a random vector with components $u,v$ 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:

:$f(u,v)\; =\; \{1\backslash over\{2\backslash pi\backslash sigma^\{2\}\}\}\backslash left(\; 1\; +\; \{u^\{2\}+v^\{2\}\backslash over\{\backslash nu\; \backslash sigma^\{2\}\}\}\; \backslash right)^\{-\backslash nu/2-1\}$

Let $R\; =\; \backslash sqrt\{U^\{2\}+V^\{2\}\}$ be the magnitude of $Y$. Then the cumulative distribution function (CDF) of the magnitude is:

:$F(r)\; =\; \{1\backslash over\{2\backslash pi\backslash sigma^\{2\}\}\}\backslash iint\_\{D\_\{r\}\}\; \backslash left(\; 1\; +\; \{u^\{2\}+v^\{2\}\backslash over\{\backslash nu\; \backslash sigma^\{2\}\}\}\; \backslash right)^\{-\backslash nu/2-1\}du\; \backslash ;\; dv$

where $D\_\{r\}$ is the disk defined by:

:$D\_\{r\}\; =\; \backslash left\backslash \{\; (u,v)\; :\; \backslash sqrt\{u^\{2\}+v^\{2\}\}\; \backslash leq\; r\; \backslash right\backslash \}$

Converting to polar coordinates leads to the CDF becoming:

:

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

:$f(r)\; =\; F\text{'}(r)\; =\; \{r\backslash over\{\backslash sigma^\{2\}\}\}\; \backslash left(\; 1\; +\; \{r^\{2\}\backslash over\{\backslash nu\; \backslash sigma^\{2\}\}\}\; \backslash right)^\{-\backslash nu/2-1\}$

In the limit as $\backslash nu\; \backslash rightarrow\; \backslash infty$, the Rayleigh distribution is recovered because:

:$\backslash lim\_\{\backslash nu\backslash rightarrow\; \backslash infty\}\; \backslash left(\; 1\; +\; \{r^\{2\}\backslash over\{\backslash nu\; \backslash sigma^\{2\}\}\}\; \backslash right)^\{-\backslash nu/2-1\}\; =\; e^\{-r^\{2\}/2\backslash sigma^\{2\}\}$

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The raw moments are given by:

: $\backslash mu\_j\; =\; \backslash sigma^j2^\{j/2\}\backslash ,\backslash Gamma\backslash left(1\; +\; \backslash frac\; j\; 2\backslash right),$

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

The mean of a Rayleigh random variable is thus :

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

The standard deviation of a Rayleigh random variable is:

:$\backslash operatorname\{std\}(X)\; =\; \backslash sqrt\{\backslash left\; (2-\backslash frac\{\backslash pi\}\{2\}\backslash right)\}\; \backslash sigma\; \backslash approx\; 0.655\backslash \; \backslash sigma$

The variance of a Rayleigh random variable is :

:$\backslash operatorname\{var\}(X)\; =\; \backslash mu\_2-\backslash mu\_1^2\; =\; \backslash left(2-\backslash frac\{\backslash pi\}\{2\}\backslash right)\; \backslash sigma^2\; \backslash approx\; 0.429\backslash \; \backslash sigma^2$

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

:$f\_\{\backslash max\}\; =\; f(\backslash sigma;\backslash sigma)\; =\; \backslash frac\{1\}\{\backslash sigma\}\; e^\{-1/2\}\; \backslash approx\; \backslash frac\{0.606\}\{\backslash sigma\}.$

The skewness is given by:

:$\backslash gamma\_1\; =\; \backslash frac\{2\backslash sqrt\{\backslash pi\}(\backslash pi\; -\; 3)\}\{(4\; -\; \backslash pi)^\{3/2\}\}\; \backslash approx\; 0.631$

The excess kurtosis is given by:

:$\backslash gamma\_2\; =\; -\backslash frac\{6\backslash pi^2\; -\; 24\backslash pi\; +\; 16\}\{(4\; -\; \backslash pi)^2\}\; \backslash approx\; 0.245$

The characteristic function is given by:

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

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

:$$

M(t) = 1 + \sigma t\,e^{\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}}

\left[\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right]

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

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

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

where $\backslash gamma$ is the Euler–Mascheroni constant.

Given a sample of N independent and identically distributed Rayleigh random variables $x\_i$ with parameter $\backslash sigma$,

: $\backslash widehat\{\backslash sigma^2\}\; =\; \backslash !\backslash ,\backslash frac\{1\}\{2N\}\backslash sum\_\{i=1\}^N\; x\_i^2$ is the maximum likelihood estimate and also is unbiased.

:$\backslash widehat\{\backslash sigma\}\backslash approx\; \backslash sqrt\{\backslash frac\; 1\; \{2N\}\; \backslash sum\_\{i=1\}^N\; x\_i^2\}$ is a biased estimator that can be corrected via the formula

:$\backslash sigma\; =\; \backslash widehat\{\backslash sigma\}\; \backslash frac\; \{\backslash Gamma(N)\backslash sqrt\{N\}\}\; \{\backslash Gamma\backslash left(N\; +\; \backslash frac\; 1\; 2\backslash right)\}\; =\; \backslash widehat\{\backslash sigma\}\; \backslash frac\; \{4^N\; N!(N-1)!\backslash sqrt\{N\}\}\; \{(2N)!\backslash sqrt\{\backslash pi\}\}$[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] $=\; \backslash frac\{\backslash hat\{\backslash sigma\}\}\{c\_4(2N+1)\}$, where c₄ is the correction factor used to unbias estimates of standard deviation for normal random variables.

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

:  $P\backslash left(\backslash chi\_\{2N\}^2\; \backslash leq\; a\backslash right)\; =\; \backslash alpha/2,\; \backslash quad\; P\backslash left(\backslash chi\_\{2N\}^2\; \backslash leq\; b\backslash right)\; =\; 1\; -\; \backslash alpha/2$

then the scale parameter will fall within the bounds

:  $\backslash frac\{\{N\}\backslash overline\{x^2\}\}\{b\}\; \backslash leq\; \{\backslash widehat\{\backslash sigma^2\}\}\; \backslash leq\; \backslash frac\{\{N\}\backslash overline\{x^2\}\}\{a\}$[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]

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

:$X=\backslash sigma\backslash sqrt\{-2\; \backslash ln\; U\}\backslash ,$

has a Rayleigh distribution with parameter $\backslash sigma$. This is obtained by applying the inverse transform sampling-method.

• $R\; \backslash sim\; \backslash mathrm\{Rayleigh\}(\backslash sigma)$ is Rayleigh distributed if $R\; =\; \backslash sqrt\{X^2\; +\; Y^2\}$, where $X\; \backslash sim\; N(0,\; \backslash sigma^2)$ and $Y\; \backslash sim\; N(0,\; \backslash sigma^2)$ are independent normal random variables.[https://web.archive.org/web/20131105232146/http://home.kpn.nl/jhhogema1966/skeetn/ballist/sgs/sgs.htm#_Toc96439743 Hogema, Jeroen (2005) "Shot group statistics"] This gives motivation to the use of the symbol $\backslash sigma$ in the above parametrization of the Rayleigh density.
• The magnitude $|z|$ of a standard complex normally distributed variable z is Rayleigh distributed.
• The chi distribution with v = 2 is equivalent to the Rayleigh Distribution with σ = 1: $R(\backslash sigma)\; \backslash sim\; \backslash sigma\backslash chi\_2^\{\backslash ,\}\backslash \; .$
• If $R\; \backslash sim\; \backslash mathrm\{Rayleigh\}\; (1)$, then $R^2$ has a chi-squared distribution with 2 degrees of freedom: $[Q=R(\backslash sigma)^2]\; \backslash sim\; \backslash sigma^2\backslash chi\_2^2\backslash \; .$
• If $R\; \backslash sim\; \backslash mathrm\{Rayleigh\}(\backslash sigma)$, then $\backslash sum\_\{i=1\}^N\; R\_i^2$ has a gamma distribution with integer scale parameter $N$ and rate parameter $\backslash frac\{1\}\{2\backslash sigma^2\}$
• : $\backslash left[Y=\backslash sum\_\{i=1\}^N\; R\_i^2\backslash right]\; \backslash sim\; \backslash Gamma\backslash left(N,\backslash frac\{1\}\{2\backslash sigma^2\}\backslash right)$ with integer shape parameter N and rate parameter $\backslash frac\{1\}\{2\backslash sigma^2\}.$
• : $\backslash left[Y=\backslash sum\_\{i=1\}^N\; R\_i^2\backslash right]\; \backslash sim\; \backslash Gamma\backslash left(N,2\backslash sigma^2\backslash right)$ with integer shape parameter N and scale parameter $2\backslash sigma^2.$
• The Rice distribution is a noncentral generalization of the Rayleigh distribution: $\backslash mathrm\{Rayleigh\}(\backslash sigma)\; =\; \backslash mathrm\{Rice\}(0,\backslash sigma)$.
• The Weibull distribution with the shape parameter k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter $\backslash sigma$ is related to the Weibull scale parameter according to $\backslash lambda\; =\; \backslash sigma\; \backslash sqrt\{2\}\; .$
• If $X$ has an exponential distribution $X\; \backslash sim\; \backslash mathrm\{Exponential\}(\backslash lambda)$, then $Y=\backslash sqrt\{X\}\; \backslash sim\; \backslash mathrm\{Rayleigh\}(1/\backslash sqrt\{2\backslash lambda\})\; .$
• 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.

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.{{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 }}

{{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 }}

The 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.{{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|pages=e0187292|doi=10.1371/journal.pone.0187292|pmid=29161271|issn=1932-6203|bibcode=2017PLoSO..1287292A|pmc=5697816|doi-access=free}}

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.{{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}}

• Circular error probable
• Rayleigh fading
• Rayleigh mixture distribution
• Rice distribution

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Category:Continuous distributions

Category:Exponential family distributions