Inprobability theory andstatistics aRayleigh mixture distribution is a weighted mixture of multipleprobability distributions where the weightings are equal to the weightings of aRayleigh distribution.^[1] Since theprobability density function for a (standard) Rayleigh distribution is given by^[2]

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

Rayleigh mixture distributions have probability density functions of the form

${\displaystyle f(x;\sigma ,n)={\int }_0^{\mathrm{\infty }}\frac{r{e}^{-r^2/2{\sigma }^2}}{{\sigma }^2}\tau (x,r;n)\mathrm{d}r,}$

where${\displaystyle \tau (x,r;n)}$ is a well-defined probability density function orsampling distribution.^[1]

The Rayleigh mixture distribution is one of many types ofcompound 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 inmixture models, which are used to express probabilities of sub-populations within a larger population.

• Mixture distribution
• List of probability distributions

• Continuous distributions
• Compound probability distributions