Rayleigh mixture distribution explained

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.^[1] Since the probability density function for a (standard) Rayleigh distribution is given by^[2]

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

Rayleigh mixture distributions have probability density functions of the form

f(x;\sigma,n) = \int ^{inf ty} _{0} (re ^{- r ^{2/2\sigma} ^{2}})/(\sigma ^{2}) \tau(x,r;n) 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

Notes and References

1. Karim R., Hossain P., Begum S., and Hossain F., "Rayleigh Mixture Distribution", Journal of Applied Mathematics, Vol. 2011, (2011).
2. Jackson J.L., "Properties of the Rayleigh Distribution", Johns Hopkins University (1954).