Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
[1] A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional components.
One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions.
Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, which is infrequent, 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.
The probability density function of the Rayleigh distribution is[2] where
which has components that are bivariate normally distributed, centered at zero, with equal variances
is the disk Writing the double integral in polar coordinates, it becomes Finally, the probability density function for
There are also generalizations when the components have unequal variance or correlations (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).
If the components both have mean zero, equal variance and are independent, the bivariate Student's-t distribution takes the form: Let
Then the cumulative distribution function (CDF) of the magnitude is: where
is the disk defined by: Converting to polar coordinates leads to the CDF becoming: Finally, the probability density function (PDF) of the magnitude may be derived: In the limit as
, the Rayleigh distribution is recovered because: The raw moments are given by: where
and the maximum pdf is The skewness is given by: The excess kurtosis is given by: The characteristic function is given by: where
Given a sample of N independent and identically distributed Rayleigh random variables
where: then the scale parameter will fall within the bounds 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.
An application of the estimation of σ can be found in magnetic resonance imaging (MRI).
Hence, the above formula can be used to estimate the noise variance in an MRI image from background data.
[7] [8] The Rayleigh distribution was also employed in the field of nutrition for linking dietary nutrient levels and human and animal responses.
In this way, the parameter σ may be used to calculate nutrient response relationship.
[9] In the field of ballistics, the Rayleigh distribution is used for calculating the circular error probable—a measure of a gun's precision.