Editing Mixture distribution (section)

== Uncountable mixtures ==
{{Main article|Compound distribution}}

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

<math display="block"> f(x) = \int_A \, w(a) \, p(x;a) \, da </math>

is again a probability density function for {{mvar|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 {{mvar|w}} is allowed to be a [[generalized function]] representing the "derivative" of the cumulative distribution function of a [[discrete distribution]].