|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 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 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.
Information entropy 
where is the Euler–Mascheroni constant.
Parameter estimation 
Given N independent and identically distributed Rayleigh random variables with parameter , the maximum likelihood estimate of is
An application of the estimation of 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.
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.
Related distributions 
- 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.)
- 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 Chi distribution with v=2 is equivalent to Rayleigh Distribution with sigma=1
- 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
See also 
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (April 2013)|
- Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processe. ISBN 0073660116, ISBN 9780073660110[page needed]
- Sijbers J., den Dekker A. J., Raman E. and Van Dyck D. (1999) "Parameter estimation from magnitude MR images", International Journal of Imaging Systems and Technology, 10(2), 109–114
A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.