Probability density function
Cumulative distribution function
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 each component is 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 independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.
Relation to random vector lengths
Consider the two-dimensional vector which has components that are Gaussian-distributed, centered at zero, and independent. Then , and similarly for .
Let be the length of . It is distributed as
By transforming to the polar coordinate system one has
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.
The raw moments are given by:
where is the Gamma function.
The mode is and the maximum pdf is
The skewness is given by:
The excess kurtosis is given by:
The characteristic function is given by:
where is the error function.
where is the Euler–Mascheroni constant.
The pdf of the Rayleigh distribution is a solution of the following differential equation:
Given a sample of N independent and identically distributed Rayleigh random variables with parameter ,
- is an unbiased maximum likelihood estimate.
- is a biased estimator that can be corrected via the formula
To find the (1 − α) confidence interval, first find the two numbers where:
Generating random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
has a Rayleigh distribution with parameter . This is obtained by applying the inverse transform sampling-method.
- is Rayleigh distributed if , where and are independent normal random variables. (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
- The chi distribution with v = 2 is equivalent to Rayleigh Distribution with σ = 1. I.e., if , then has a chi-squared distribution with parameter , degrees of freedom, equal to two (N = 2)
- If , then has a gamma distribution with parameters and
- The Rice distribution is a generalization of the Rayleigh distribution.
- The Weibull distribution is a generalization of the Rayleigh distribution. In this instance, parameter is related to the Weibull scale parameter :
- The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
- If has an exponential distribution , then
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. 
Proof of correctness – Unequal variances
||This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (April 2013)|
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