Rayleigh mixture distribution

Not to be confused with Rayleigh scattering.

In probability theory and statistics a Rayleigh mixture distribution is a weighted mixture of multiple probability distributions where the weightings are equal to the weightings of a Rayleigh distribution. Since the probability density function for a (standard) Rayleigh distribution is given by

$f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/2\sigma ^{2}},\quad x\geq 0,$

Rayleigh mixture distributions have probability density functions of the form

$f(x;\sigma ,n)=\int _{0}^{\infty }{\frac {re^{-r^{2}/2\sigma ^{2}}}{\sigma ^{2}}}\tau (x,r;n)\,\mathrm {d} r,$

where $\tau (x,r;n)$ is a well-defined probability density function or sampling distribution.

The Rayleigh mixture distribution is one of many types of compound distributions in which the appearance of a value in a sample or population might be interpreted as a function of other underlying random variables. Mixture distributions are often used in mixture models, which are used to express probabilities of sub-populations within a larger population.

See also

• Mixture distribution
• List of probability distributions