Rayleigh mixture distribution

{{Distinguish|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.Karim R., Hossain P., Begum S., and Hossain F., "Rayleigh Mixture Distribution", Journal of Applied Mathematics, Vol. 2011, {{doi|10.1155/2011/238290}} (2011). Since the probability density function for a (standard) Rayleigh distribution is given byJackson J.L., "Properties of the Rayleigh Distribution", Johns Hopkins University (1954).

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

Rayleigh mixture distributions have probability density functions of the form

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

where $\backslash tau(x,r;n)$ 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.