Rayleigh distribution

Probability density function as a function of ${\displaystyle \sigma }$

In probability theory and statistics , the Rayleigh distribution (after John William Strutt, 3rd Baron Rayleigh ) denotes a continuous probability distribution .

If the components of a two-dimensional random vector are normally distributed and stochastically independent , then the amount is Rayleigh distributed. This occurs, for example, in a transmission channel used for radio technology in mobile radio systems when there is no direct visual contact between the transmitter, such as a base station, and the receiver, for example a mobile phone. The transmission channel impaired by the multipath propagation via various, random reflections and scattering, for example on building walls and other obstacles, can then be modeled as a so-called Rayleigh channel with the help of the Rayleigh distribution .

The distribution of 10-minute mean values ​​of wind speed is also often described by a Rayleigh distribution, unless a two-parameter Weibull distribution is to be selected.

definition

A continuous random variable is called Rayleigh distributed with parameters if it has the probability density ${\displaystyle X}$ ${\displaystyle \sigma >0}$

${\displaystyle f(x|\sigma )={\begin{cases}\displaystyle {\frac {x}{\sigma ^{2}}}e^{-{\frac {x^{2}}{2\sigma ^{2}}}}&x\geq 0\\0&x<0\end{cases}}}$

owns. This gives the distribution function

${\displaystyle F(x)={\begin{cases}\displaystyle 1-e^{-{\frac {x^{2}}{2\sigma ^{2}}}}&x\geq 0\\0&x<0\end{cases}}}$

properties

Moments

The moments of any order can be calculated using the following formula:

${\displaystyle \mu _{k}=\sigma ^{k}2^{k/2}\,\Gamma (1+k/2)\,}$,

where represents the gamma function . ${\displaystyle \Gamma (\cdot )}$

Expected value

The expected value results in

${\displaystyle \operatorname {E} (X)=\sigma {\sqrt {\frac {\pi }{2}}}}$.

Variance

The variance of the distribution is

${\displaystyle \operatorname {Var} (X)={\frac {4-\pi }{2}}\sigma ^{2}}$.

Thus, the relationship between the expected value and the standard deviation is constant for this distribution:

${\displaystyle {\frac {\operatorname {E} (X)}{\sqrt {\operatorname {Var} (X)}}}={\sqrt {\frac {\pi }{2}}}{\sqrt {\frac {2}{4-\pi }}}={\sqrt {\frac {\pi }{4-\pi }}}\approx 1{,}91}$.

Crookedness

For the skew you get

${\displaystyle \operatorname {v} (X)={\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}\approx 0{,}6311}$.

Bulge (kurtosis)

The bulge arises too

${\displaystyle \beta _{2}(X)=-{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}\approx 0{,}2451}$.

Characteristic function

The characteristic function is

${\displaystyle \varphi (t)=1-\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\!\left(\operatorname {erf} \!\left({\frac {\sigma t}{\sqrt {2}}}\right)-i\right)}$.

where is the complex error function . ${\displaystyle \operatorname {erf} (\cdot )}$

Moment generating function

The torque generating function is given by

${\displaystyle M(t)=1+\sigma te^{\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left(\operatorname {erf} \left({\frac {\sigma t}{\sqrt {2}}}\right)+1\right)}$,

where again is the error function. ${\displaystyle \operatorname {erf} (\cdot )}$

entropy

The entropy , expressed in nats , results in

${\displaystyle 1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}}$,

where denotes the Euler-Mascheroni constant . ${\displaystyle \gamma }$

mode

The Rayleigh distribution reaches the maximum for , because for holds ${\displaystyle x=\sigma }$ ${\displaystyle x\geq 0}$

${\displaystyle 0={\frac {\mathrm {d} f}{\mathrm {d} x}}\left(x\right)={\frac {e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}{\sigma ^{2}}}-x^{2}{\frac {e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}{\sigma ^{4}}}\quad \Longleftrightarrow \quad x=\sigma }$.

This is the mode of the Rayleigh distribution. ${\displaystyle \sigma }$

In the maximum has the value ${\displaystyle f}$

${\displaystyle f\left(\sigma \right)={\frac {1}{\sigma }}e^{-{\frac {1}{2}}}}$.

Parameter estimation

The maximum likelihood estimation of measured values is carried out using: ${\displaystyle \sigma }$ ${\displaystyle x_{1},\ldots ,x_{n}}$

${\displaystyle \sigma \approx {\sqrt {{\frac {1}{2n}}\sum _{i=1}^{n}x_{i}^{2}}}}$

Relationships with other distributions

The Chi distribution , Weibull distribution, and Rice distribution are generalizations of the Rayleigh distribution.

Relationship to the chi-square distribution

If , then chi-square is distributed with two degrees of freedom : ${\displaystyle R\sim \mathrm {Rayleigh} (1)}$ ${\displaystyle R^{2}}$ ${\displaystyle R^{2}\sim \chi _{2}^{2}}$

Relationship to the Weibull distribution

${\displaystyle \mathrm {Rayleigh} (\sigma ^{2})=\mathrm {Wei} \left({\frac {1}{2\sigma ^{2}}},2\right)}$

Relationship to Rice Distribution

${\displaystyle \mathrm {Rayleigh} (\sigma )=\mathrm {Rice} (0,\sigma )}$

Relationship to the exponential distribution

If exponentially is , then . ${\displaystyle X}$ ${\displaystyle \!\,X\sim \mathrm {Exp} (\lambda )}$ ${\displaystyle Y={\sqrt {X}}\sim \mathrm {Rayleigh} \left({\frac {1}{\sqrt {2\lambda }}}\right)}$

Relationship to the gamma distribution

If , then gamma distributed with parameters and : . ${\displaystyle R\sim \mathrm {Rayleigh} (\sigma )}$ ${\displaystyle \sum _{i=1}^{N}R_{i}^{2}}$ ${\displaystyle N}$ ${\displaystyle 2\sigma ^{2}}$ ${\displaystyle Y=\sum _{i=1}^{N}R_{i}^{2}\sim \Gamma (N,2\sigma ^{2})}$

Relationship to normal distribution

${\displaystyle {\sqrt {X^{2}+Y^{2}}}}$is Rayleigh distributed if and are two stochastically independent normally distributed random variables. ${\displaystyle X\sim {\mathcal {N}}(0,\sigma ^{2})}$ ${\displaystyle Y\sim {\mathcal {N}}(0,\sigma ^{2})}$

literature

• Edgar Dietrich, Alfred Schulze: Statistical procedures for machine and process qualification . 6th edition. Carl Hanser Verlag, 2009, ISBN 978-3-446-41525-6 .