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Concept
Rayleigh distribution
Summary
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distribution is named after Lord Rayleigh (ˈreɪli).
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, 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
where is the scale parameter of the distribution. The cumulative distribution function is
for
Consider the two-dimensional vector which has components that are bivariate normally distributed, centered at zero, and independent. Then and have density functions
Let be the length of . That is, Then has cumulative distribution function
where is the disk
Writing the double integral in polar coordinates, it becomes
Finally, the probability density function for is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is
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 (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).
Suppose is a random vector with components that follows a multivariate t-distribution.
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