Rayleigh distribution

Not to be confused with Rayleigh mixture distribution.

Rayleigh

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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Q(F;\sigma)=\sigma \sqrt{-2\ln(1 - F)}}
Mean	$\sigma {\sqrt {\frac {\pi }{2}}}$
Median	$\sigma {\sqrt {2\ln(2)}}$
Mode	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma}
Variance	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{4 - \pi}{2} \sigma^2}
Skewness	${\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}$
Excess kurtosis	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle -\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}}
Entropy	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}}
MGF	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1+\sigma te^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right)}
CF	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1 - \sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\operatorname{erfi} \left(\frac{\sigma t}{\sqrt{2}}\right) - i\right)}

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh (/ˈreɪli/).

A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane 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, which is infrequent, 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.

Definition

The probability density function of the Rayleigh distribution is

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma} is the scale parameter of the distribution. The cumulative distribution function is

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

for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x \in [0,\infty).}

Relation to random vector length

Consider the two-dimensional vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y = (U,V) } which has components that are bivariate normally distributed, centered at zero, with equal variances Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma^2} , and independent. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle V} have density functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y} . That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X = \sqrt{U^2 + V^2}.} Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X} has cumulative distribution function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle F_X(x; \sigma) = \iint_{D_x} f_U(u;\sigma) f_V(v;\sigma) \,dA,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle D_x} is the disk

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle D_x = \left\{(u,v) : \sqrt{u^2 + v^2} \leq x\right\}.}

Writing the double integral in polar coordinates, it becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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. }

Finally, the probability density function for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X} is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).

Generalization to bivariate Student's t-distribution

Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y} is a random vector with components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f(u,v) = {1\over{2\pi\sigma^{2}}}\left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}} Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R = \sqrt{U^{2}+V^{2}}} be the magnitude of $Y$. Then the cumulative distribution function (CDF) of the magnitude is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle D_{r}} is the disk defined by: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle D_{r} = \left\{ (u,v) : \sqrt{u^{2}+v^{2}} \leq r \right\} } Converting to polar coordinates leads to the CDF becoming: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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} } Finally, the probability density function (PDF) of the magnitude may be derived: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f(r) = F'(r) = {r\over{\sigma^{2}}} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} } In the limit as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \nu \rightarrow \infty } , the Rayleigh distribution is recovered because: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \lim_{\nu\rightarrow \infty} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} = e^{-r^{2}/2\sigma^{2}} }

Properties

The raw moments are given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mu_j = \sigma^j2^{j/2}\,\Gamma\left(1 + \frac j 2\right),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Gamma(z)} is the gamma function.

The mean of a Rayleigh random variable is thus :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253\ \sigma.}

The standard deviation of a Rayleigh random variable is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \operatorname{std}(X) = \sqrt{\left (2-\frac{\pi}{2}\right)} \sigma \approx 0.655\ \sigma}

The variance of a Rayleigh random variable is :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \operatorname{var}(X) = \mu_2-\mu_1^2 = \left(2-\frac{\pi}{2}\right) \sigma^2 \approx 0.429\ \sigma^2}

The mode is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma,} and the maximum pdf is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f_{\max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-1/2} \approx \frac{0.606}{\sigma}.}

The skewness is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^{3/2}} \approx 0.631}

The excess kurtosis is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245}

The characteristic function is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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]}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \operatorname{erfi}(z)} is the imaginary error function. The moment generating function is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \operatorname{erf}(z)} is the error function.

Differential entropy

The differential entropy is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle H = 1 + \ln\left(\frac \sigma {\sqrt{2}}\right) + \frac \gamma 2 }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \gamma} is the Euler–Mascheroni constant.

Parameter estimation

Given a sample of N independent and identically distributed Rayleigh random variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x_i} with parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \widehat{\sigma^2} = \!\,\frac{1}{2N}\sum_{i=1}^N x_i^2} is the maximum likelihood estimate and also is unbiased.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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}}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle = \frac{\hat{\sigma}}{c_4(2N+1)}} , where c₄ is the correction factor used to unbias estimates of standard deviation for normal random variables.

Confidence intervals

To find the (1 − α) confidence interval, first find the bounds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle [a,b]} where:

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P\left(\chi_{2N}^2 \leq a\right) = \alpha/2, \quad P\left(\chi_{2N}^2 \leq b\right) = 1 - \alpha/2}

then the scale parameter will fall within the bounds

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{{N}\overline{x^2}}{b} \leq {\widehat{\sigma^2}} \leq \frac{{N}\overline{x^2}}{a}}

Generating random variates

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X=\sigma\sqrt{-2 \ln U}\,}

has a Rayleigh distribution with parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma} . This is obtained by applying the inverse transform sampling-method.

Related distributions

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R \sim \mathrm{Rayleigh}(\sigma)} is Rayleigh distributed if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R = \sqrt{X^2 + Y^2}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X \sim N(0, \sigma^2)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y \sim N(0, \sigma^2)} are independent normal random variables. This gives motivation to the use of the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma} in the above parametrization of the Rayleigh density.
• The magnitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle |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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R(\sigma) \sim \sigma\chi_2^{\,}\ .}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R \sim \mathrm{Rayleigh} (1)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R^2} has a chi-squared distribution with 2 degrees of freedom: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle [Q=R(\sigma)^2] \sim \sigma^2\chi_2^2\ .}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R \sim \mathrm{Rayleigh}(\sigma)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sum_{i=1}^N R_i^2} has a gamma distribution with integer scale parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle N} and rate parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{1}{2\sigma^2}}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,\frac{1}{2\sigma^2}\right)} with integer shape parameter N and rate parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{1}{2\sigma^2}.}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,2\sigma^2\right)} with integer shape parameter N and scale parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 2\sigma^2.}

• The Rice distribution is a noncentral generalization of the Rayleigh distribution: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sigma} is related to the Weibull scale parameter according to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \lambda = \sigma \sqrt{2} .}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X} has an exponential distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X \sim \mathrm{Exponential}(\lambda)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y=\sqrt{X} \sim \mathrm{Rayleigh}(1/\sqrt{2\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.

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.

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.

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.

See also

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