Editing Mixture distribution (section)

== Finite and countable mixtures ==
[[Image:Gaussian-mixture-example.svg|thumb|Density of a mixture of three normal distributions ({{math|1=''μ'' = 5, 10, 15}}, {{math|1=''σ'' = 2}}) with equal weights. Each component is shown as a weighted density (each integrating to 1/3)]]

Given a finite set of probability density functions {{math|''p''<sub>1</sub>(''x'')}}, ..., {{math|''p<sub>n</sub>''(''x'')}}, or corresponding cumulative distribution functions {{math|''P''<sub>1</sub>(''x''),}} ..., {{math|''P<sub>n</sub>''(''x'')}} and '''weights''' {{math|''w''<sub>1</sub>}}, ..., {{math|''w<sub>n</sub>''}} such that {{math|''w<sub>i</sub>'' ≥ 0}} and {{math|1=∑''w<sub>i</sub>'' = 1}}, the mixture distribution can be represented by writing either the density, {{math|''f''}}, or the distribution function, {{math|''F''}}, as a sum (which in both cases is a convex combination):
<math display="block"> F(x) = \sum_{i=1}^n \, w_i \, P_i(x), </math>
<math display="block"> f(x) = \sum_{i=1}^n \, w_i \, p_i(x) .</math>
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 <math> n = \infty\!</math>.