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]

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

Rayleigh mixture distributions have probability density functions of the form

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

where ${\displaystyle \tau (x,r;n)}$ is a well-defined probability density function 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.