Rayleigh distribution

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Rayleigh
Probability density function
Plot of the Rayleigh PDF
Cumulative distribution function
Plot of the Rayleigh CDF
Parameters \sigma>0\,
Support x\in [0,+\infty)
pdf \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}
CDF 1 - e^{-x^2/2\sigma^2}
Mean \sigma \sqrt{\frac{\pi}{2}}
Median \sigma\sqrt{\ln(4)}\,
Mode \sigma\,
Variance \frac{4 - \pi}{2} \sigma^2
Skewness \frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}
Ex. kurtosis -\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}
Entropy 1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}
MGF 1+\sigma t\,e^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}
\left(\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right)
CF 1\!-\!\sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}\!\left(\textrm{erfi}\!\left(\frac{\sigma t}{\sqrt{2}}\right)\!-\!i\right)

In probability theory and statistics, the Rayleigh distribution /ˈrli/ is a continuous probability distribution for positive-valued random variables.

A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed into its orthogonal 2-dimensional vector components. Assuming that the magnitudes of each component are uncorrelated, normally distributed with equal variance, and zero mean, 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 i.i.d. (independently and identically distributed) Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.

The distribution is named after Lord Rayleigh.[citation needed]

Definition[edit]

The probability density function of the Rayleigh distribution is[1]

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

where \sigma >0, is the scale parameter of the distribution. The cumulative distribution function is[1]

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

for x \in [0,\infty].

Relation to random vector lengths[edit]

Consider the two-dimensional vector  Y = (U,V) which has components that are Gaussian-distributed and independent. Then  f_U(u; \sigma) = \frac{e^{-u^2/2\sigma^2}}{\sqrt{2\pi\sigma^2}} , and similarly for  f_V(v; \sigma) .

Let  x be the length of  Y . It is distributed as

f(x; \sigma) =  \frac{1}{2\pi\sigma^2} \int_{-\infty}^\infty du \, \int_{-\infty}^\infty dv \, e^{-u^2/2\sigma^2} e^{-v^2/2\sigma^2} \delta(x-\sqrt{u^2+v^2}).

By transforming to the polar coordinate system one has

 f(x; \sigma) = \frac{1}{2\pi\sigma^2} \int_0^{2\pi} \, d\phi \int_0^\infty dr \, \delta(r-x) r e^{-r^2/2\sigma^2}= \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.

Properties[edit]

The raw moments are given by:

\mu_k = \sigma^k2^\frac{k}{2}\,\Gamma\left(1 + \frac{k}{2}\right)

where \Gamma(z) is the Gamma function.

The mean and variance of a Rayleigh random variable may be expressed as:

\mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253 \sigma

and

\textrm{var}(X) = \frac{4 - \pi}{2} \sigma^2 \approx 0.429 \sigma^2

The mode is \sigma and the maximum pdf is

 f_\text{max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-\frac{1}{2}} \approx \frac{1}{\sigma} 0.606

The skewness is given by:

\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^\frac{3}{2}} \approx 0.631

The excess kurtosis is given by:

\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245

The characteristic function is given by:

\varphi(t) = 1 - \sigma te^{-\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\textrm{erfi} \left(\frac{\sigma t}{\sqrt{2}}\right) - i\right]

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

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

Differential entropy[edit]

The differential entropy is given by[citation needed]

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

where \gamma is the Euler–Mascheroni constant.


Differential equation[edit]


\left\{\sigma ^2 x f'(x)+f(x) \left(x^2-\sigma
   ^2\right)=0,f(1)=\frac{e^{-\frac{1}{2 \sigma ^2}}}{\sigma ^2}\right\}

Parameter estimation[edit]

Given a sample of N independent and identically distributed Rayleigh random variables x_i with parameter \sigma,

\widehat{\sigma^2}\approx \!\,\frac{1}{2N}\sum_{i=1}^N x_i^2 is an unbiased maximum likelihood estimate.
\hat{\sigma}\approx \!\,\sqrt{\frac{1}{2N}\sum_{i=1}^N x_i^2} is a biased estimator that can be corrected via the formula
\sigma = \hat{\sigma} \frac {\Gamma(N)\sqrt{N}} {\Gamma(N + \frac {1} {2})} = \hat{\sigma} \frac {4^N N!(N-1)!\sqrt{N}} {(2N)!\sqrt{\pi}}[2]

Confidence intervals[edit]

To find the (1 − α) confidence interval, first find the two numbers \chi_1^2, \ \chi_2^2 where:

  Pr(\chi^2(2N) \leq \chi_1^2) = \alpha/2, \quad Pr(\chi^2(2N) \leq \chi_2^2) = 1 - \alpha/2

then

  \frac{N\overline{x^2}}{\chi_2^2} \leq \widehat{\sigma}^2 \leq \frac{N\overline{x^2}}{\chi_1^2}[3]

Generating random variates[edit]

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

X=\sigma\sqrt{-2 \ln(U)}\,

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

Related distributions[edit]

  • R \sim \mathrm{Rayleigh}(\sigma) is Rayleigh distributed if R = \sqrt{X^2 + Y^2}, where X \sim N(0, \sigma^2) and Y \sim N(0, \sigma^2) are independent normal random variables.[4] (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
[Q=R^2] \sim \chi^2(N)\ .
\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma(N,2\sigma^2) .
  • The Weibull distribution is a generalization of the Rayleigh distribution. In this instance, parameter \sigma is related to the Weibull scale parameter \lambda: \lambda = \sigma \sqrt{2} .

Applications[edit]

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.[5]

See also[edit]

References[edit]