| median =$\backslash sigma\backslash sqrt\{\backslash ln(4)\}\backslash ,$| mode =$\backslash sigma\backslash ,$| variance =$\backslash frac\{4\; -\; \backslash pi\}\{2\}\; \backslash sigma^2$| skewness =$\backslash frac\{2\backslash sqrt\{\backslash pi\}(\backslash pi\; -\; 3)\}\{(4-\backslash pi)^\{3/2\}\}$| kurtosis =$-\backslash frac\{6\backslash pi^2\; -\; 24\backslash pi\; +16\}\{(4-\backslash pi)^2\}$| entropy =$1+\backslash ln\backslash left(\backslash frac\{1\}\{\backslash sqrt\{2\}\backslash sigma^3\}\backslash right)+\backslash frac\{\backslash gamma\}\{2\}$| mgf =$1+\backslash sigma\; t\backslash ,e^\{\backslash sigma^2t^2/2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash left(\backslash textrm\{erf\}\backslash left(\backslash frac\{\backslash sigma\; t\}\{\backslash sqrt\{2\}\}\backslash right)\backslash !+\backslash !1\backslash right)$| char =$1\backslash !-\backslash !\backslash sigma\; te^\{-\backslash sigma^2t^2/2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash !\backslash left(\backslash textrm\{erfi\}\backslash !\backslash left(\backslash frac\{\backslash sigma\; t\}\{\backslash sqrt\{2\}\}\backslash right)\backslash !-\backslash !i\backslash right)$| }}

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. It usually arises when a two dimensional vector (e.g. wind velocity) has its two orthogonal components normally and independently distributed. The absolute value (e.g. wind speed) will then have a Rayleigh distribution. The distribution may also arise in the case of random complex numbers whose real and imaginary components are normally and independently distributed. The absolute value of these numbers will then be Rayleigh distributed.

The probability density function is

$f(x|\backslash sigma)\; =\; \backslash frac\{x\; \backslash exp\backslash left(\backslash frac\{-x^2\}\{2\backslash sigma^2\}\backslash right)\}\{\backslash sigma^2\}$

The characteristic function is given by:

$\backslash varphi(t)=$

$1\backslash !-\backslash !\backslash sigma\; te^\{-\backslash sigma^2t^2/2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash !\backslash left(\backslash textrm\{erfi\}\backslash !\backslash left(\backslash frac\{\backslash sigma\; t\}\{\backslash sqrt\{2\}\}\backslash right)\backslash !-\backslash !i\backslash right)$

where $erfi(z)$ is the complex error function. The moment generating function is given by:

$M(t)=\backslash ,$

$1+\backslash sigma\; t\backslash ,e^\{\backslash sigma^2t^2/2\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}$

\left(\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right)

where $erf(z)$ is the error function. The raw moments are then given by:

$\backslash mu\_k=\backslash sigma^k2^\{k/2\}\backslash ,\backslash Gamma(1+k/2)\backslash ,$

where $\backslash Gamma(z)$ is the Gamma function. The moments may be used to calculate:

Mean: $\backslash sigma\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}$

Variance: $\backslash frac\{4-\backslash pi\}\{2\}\; \backslash sigma^2$

Skewness: $\backslash frac\{2\backslash sqrt\{\backslash pi\}(\backslash pi\; -\; 3)\}\{(4-\backslash pi)^\{3/2\}\}$

Kurtosis: $-\; \backslash frac\{6\backslash pi^2\; -\; 24\backslash pi\; +16\}\{(4-\backslash pi)^2\}$

Parameter estimation

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The maximum likelihood estimate of the $\backslash sigma$ parameter is given by:

$\backslash sigma\backslash approx\backslash sqrt\{\backslash frac\{1\}\{2N\}\backslash sum\_\{i=0\}^N\; x\_i^2\}$

Related distributions

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• $R\; \backslash sim\; \backslash mathrm\{Rayleigh\}(\backslash sigma)$ is a Rayleigh distribution 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 two independent normal distributions. (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
• If $R\; \backslash sim\; \backslash mathrm\{Rayleigh\}(1)$ then $R^2$ has a chi-square distribution with two degrees of freedom: $R^2\; \backslash sim\; \backslash chi^2\_2$
• If $X$ has an exponential distribution $X\; \backslash sim\; \backslash mathrm\{Exponential\}(x|\backslash lambda)$ then $Y=\backslash sqrt\{2X\backslash sigma\backslash lambda\}\; \backslash sim\; \backslash mathrm\{Rayleigh\}(y|\backslash sigma)$.

• If $R\; \backslash sim\; \backslash mathrm\{Rayleigh\}(\backslash sigma^2)$ then $\backslash sum\_\{i=1\}^N\; R\_i^2$ has a gamma distribution with parameters $N$ and $2\backslash sigma^2$: $R\_i^2\backslash sim\; \backslash Gamma(N,2\backslash sigma^2)$.

• The Chi distribution is a generalization of the Rayleigh distribution.
• The Rice distribution is a generalization of the Rayleigh distribution.
• The Weibull distribution is a generalization of the Rayleigh distribution.

See also

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• Rayleigh fading

Continuous distributions

Rayleighverteilung | Distribución de Rayleigh | Variabile casuale di Rayleigh | 레일리 분포