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 parametrized distribution, with (some of) the parameters of that distribution themselves being random variables.

The unconditional compound distribution is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the conditional distribution.

A compound probability distribution is the probability distribution that results from assuming that a random variable ${\displaystyle X}$ is distributed according to some parametrized distribution ${\displaystyle F}$ with an unknown parameter ${\displaystyle \theta }$ that is again distributed according to some other distribution ${\displaystyle G}$. The resulting distribution ${\displaystyle H}$ is said to be the distribution that results from compounding ${\displaystyle F}$ with ${\displaystyle G}$. The parameter's distribution ${\displaystyle G}$ is also called the mixing distribution or latent distribution. Technically, the unconditional distribution ${\displaystyle H}$ results from marginalizing over ${\displaystyle G}$, i.e., from integrating out the unknown parameter(s) ${\displaystyle \theta }$. Its probability density function is given by:

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