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 according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution.

In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions. The individual distributions that are combined to form the mixture distribution are called the mixture components, and the probabilities (or weights) associated with each component are called the mixture weights. The number of components in a mixture distribution is often restricted to being finite, although in some cases the components may be countably infinite in number. More general cases (i.e. an uncountable set of component distributions), as well as the countable case, are treated under the title of compound distributions.

A distinction needs to be made between a random variable whose distribution function or density is the sum of a set of components (i.e. a mixture distribution) and a random variable whose value is the sum of the values of two or more underlying random variables, in which case the distribution is given by the convolution operator. As an example, the sum of two jointly normally distributed random variables, each with different means, will still have a normal distribution. On the other hand, a mixture density created as a mixture of two normal distributions with different means will have two peaks provided that the two means are far enough apart, showing that this distribution is radically different from a normal distribution.

Mixture distributions arise in many contexts in the literature and arise naturally where a statistical population contains two or more subpopulations. They are also sometimes used as a means of representing non-normal distributions. Data analysis concerning statistical models involving mixture distributions is discussed under the title of mixture models, while the present article concentrates on simple probabilistic and statistical properties of mixture distributions and how these relate to properties of the underlying distributions.

Finite and countable mixtures

Given a finite set of probability density functions p₁(x), ..., p_n(x), or corresponding cumulative distribution functions P₁(x), ..., P_n(x) and weights w₁, ..., w_n such that w_i ≥ 0 and Σw_i = 1, the mixture distribution can be represented by writing either the density, f, or the distribution function, F, as a sum (which in both cases is a convex combination):

${\displaystyle F(x)=\sum _{i=1}^{n}\,w_{i}\,P_{i}(x),}$

${\displaystyle f(x)=\sum _{i=1}^{n}\,w_{i}\,p_{i}(x).}$

This type of mixture, being a finite sum, is called a finite mixture, and in applications, an unqualified reference to a "mixture density" usually means a finite mixture. The case of a countably infinite set of components is covered formally by allowing ${\displaystyle n=\infty \!}$ .

Uncountable mixtures

Where the set of component distributions is uncountable, the result is often called a compound probability distribution. The construction of such distributions has a formal similarity to that of mixture distributions, with either infinite summations or integrals replacing the finite summations used for finite mixtures.

Consider a probability density function p(x;a) for a variable x, parameterized by a. That is, for each value of a in some set A, p(x;a) is a probability density function with respect to x. Given a probability density function w (meaning that w is nonnegative and integrates to 1), the function

${\displaystyle f(x)=\int _{A}\,w(a)\,p(x;a)\,da}$

is again a probability density function for x. A similar integral can be written for the cumulative distribution function. Note that the formulae here reduce to the case of a finite or infinite mixture if the density w is allowed to be a generalized function representing the "derivative" of the cumulative distribution function of a discrete distribution.

Mixtures within a parametric family

The mixture components are often not arbitrary probability distributions, but instead are members of a parametric family (such as normal distributions), with different values for a parameter or parameters. In such cases, assuming that it exists, the density can be written in the form of a sum as:

${\displaystyle f(x;a_{1},\ldots ,a_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i})}$

for one parameter, or

${\displaystyle f(x;a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n})=\sum _{i=1}^{n}\,w_{i}\,p(x;a_{i},b_{i})}$

for two parameters, and so forth.

Properties

Convexity

A general linear combination of probability density functions is not necessarily a probability density, since it may be negative or it may integrate to something other than 1. However, a convex combination of probability density functions preserves both of these properties (non-negativity and integrating to 1), and thus mixture densities are themselves probability density functions.

Moments

Let X₁, ..., X_n denote random variables from the n component distributions, and let X denote a random variable from the mixture distribution. Then, for any function H(·) for which ${\displaystyle \operatorname {E} }$ exists, and assuming that the component densities p_i(x) exist,

${\displaystyle {\begin{aligned}\operatorname {E} &=\int _{-\infty }^{\infty }H(x)\sum _{i=1}^{n}w_{i}p_{i}(x)\,dx\\&=\sum _{i=1}^{n}w_{i}\int _{-\infty }^{\infty }p_{i}(x)H(x)\,dx=\sum _{i=1}^{n}w_{i}\operatorname {E} .\end{aligned}}}$

The jth moment about zero (i.e. choosing H(x) = x^j) is simply a weighted average of the jth moments of the components. Moments about the mean H(x) = (x − μ)^j involve a binomial expansion:^[1]

${\displaystyle {\begin{aligned}\operatorname {E} (X-\mu )^{j}&=\sum _{i=1}^{n}w_{i}\operatorname {E} (X_{i}-\mu _{i}+\mu _{i}-\mu )^{j}\\&=\sum _{i=1}^{n}w_{i}\sum _{k=0}^{j}\left({\begin{array}{c}j\\k\end{array}}\right)(\mu _{i}-\mu )^{j-k}\operatorname {E} (X_{i}-\mu _{i})^{k},\end{aligned}}}$

where μ_i denotes the mean of the ith component.

In the case of a mixture of one-dimensional distributions with weights w_i, means μ_i and variances σ_i², the total mean and variance will be:

${\displaystyle \operatorname {E} X=\mu =\sum _{i=1}^{n}w_{i}\mu _{i},}$

Zdroj:https://en.wikipedia.org?pojem=Mixture_distribution
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