In probability theory and statistics, the Rayleigh distribution 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 each component is uncorrelated,
normally distributed with equal variance, and zero mean, then the overall wind speed
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. 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.
Definition The probability density function of the Rayleigh
distribution is
where is the scale parameter of the distribution. The cumulative distribution function is
for 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. Properties
The raw moments are given by:
where is the Gamma function. The mean and variance of a Rayleigh random
variable may be expressed as:
and
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 imaginary error function. The moment generating function is given by
where is the error function. Differential entropy
The differential entropy is given by
where is the Euler–Mascheroni constant. Differential equation
Parameter estimation 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
Confidence intervals To find the confidence interval, first find
the two numbers where:
then
Generating random variates Given a random variate U drawn from the uniform
distribution in the interval, 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.
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
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
Applications An application of the estimation of σ can
be found in magnetic resonance imaging. 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. See also
Normal distribution Rayleigh fading
Rayleigh mixture distribution References