Rayleigh mixture distribution
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]
- [math]\displaystyle{ f(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}, \quad x \geq 0, }[/math]
Rayleigh mixture distributions have probability density functions of the form
- [math]\displaystyle{ f(x;\sigma,n) = \int_0^{\infty} \frac{re^{-r^2/2\sigma^2}}{\sigma^2} \tau(x,r;n) \,\mathrm{d}r, }[/math]
where [math]\displaystyle{ \tau(x,r;n) }[/math] is a well-defined probability density function or sampling distribution.[1]
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
References
- ↑ 1.0 1.1 Karim R., Hossain P., Begum S., and Hossain F., "Rayleigh Mixture Distribution", Journal of Applied Mathematics, Vol. 2011, doi:10.1155/2011/238290 (2011).
- ↑ Jackson J.L., "Properties of the Rayleigh Distribution", Johns Hopkins University (1954).
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