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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and statistics a Rayleigh mixture distribution is a weighted mixture of multiple
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s where the weightings are equal to the weightings of a
Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribu ...
.Karim R., Hossain P., Begum S., and Hossain F., "Rayleigh Mixture Distribution", Journal of Applied Mathematics, Vol. 2011, (2011). Since the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
for a (standard) Rayleigh distribution is given byJackson J.L., "Properties of the Rayleigh Distribution", Johns Hopkins University (1954). :f(x;\sigma) = \frac e^, \quad x \geq 0, Rayleigh mixture distributions have probability density functions of the form :f(x;\sigma,n) = \int_0^ \frac \tau(x,r;n) \,\mathrmr, where \tau(x,r;n) is a well-defined probability density function or
sampling distribution In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were se ...
. The Rayleigh mixture distribution is one of many types of
compound distribution In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some ...
s 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 In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observati ...
, which are used to express probabilities of sub-populations within a larger population.


See also

*
Mixture distribution In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection ...
*
List of probability distributions Many probability distributions that are important in theory or applications have been given specific names. Discrete distributions With finite support * The Bernoulli distribution, which takes value 1 with probability ''p'' and value 0 with pr ...


References

{{Reflist Continuous distributions Compound probability distributions