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. If the parameter is a scale parameter, the resulting mixture is also called a scale mixture.

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

Definition

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:

${\displaystyle p_{H}(x)={\displaystyle \int \limits p_{F}(x|\theta )\,p_{G}(\theta )\operatorname {d} \!\theta }}$

The same formula applies analogously if some or all of the variables are vectors.

From the above formula, one can see that a compound distribution essentially is a special case of a marginal distribution: The joint distribution of ${\displaystyle x}$ and ${\displaystyle \theta }$ is given by ${\displaystyle p(x,\theta )=p(x|\theta )p(\theta )}$, and the compound results as its marginal distribution: ${\displaystyle {\textstyle p(x)=\int p(x,\theta )\operatorname {d} \!\theta }}$. If the domain of ${\displaystyle \theta }$ is discrete, then the distribution is again a special case of a mixture distribution.

Properties

General

The compound distribution ${\displaystyle H}$ will depend on the specific expression of each distribution, as well as which parameter of ${\displaystyle F}$ is distributed according to the distribution ${\displaystyle G}$, and the parameters of ${\displaystyle H}$ will include any parameters of ${\displaystyle G}$ that are not marginalized, or integrated, out. The support of ${\displaystyle H}$ is the same as that of ${\displaystyle F}$, and if the latter is a two-parameter distribution parameterized with the mean and variance, some general properties exist.

Mean and variance

The compound distribution's first two moments are given by the law of total expectation and the law of total variance:

$${\displaystyle \operatorname {E} _{H}X=\operatorname {E} _{G}{\bigl }\operatorname {E} _{F}X|\theta {\bigr }}$$

$${\displaystyle \operatorname {Var} _{H}(X)=\operatorname {E} _{G}{\bigl }\operatorname {Var} _{F}(X|\theta ){\bigr }+\operatorname {Var} _{G}{\bigl (}\operatorname {E} _{F}X|\theta {\bigr )}}$$

If the mean of ${\displaystyle F}$ is distributed as ${\displaystyle G}$, which in turn has mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$ the expressions above imply ${\displaystyle \operatorname {E} _{H}X=\operatorname {E} _{G}\theta =\mu }$ and ${\displaystyle \operatorname {Var} _{H}(X)=\operatorname {Var} _{F}(X|\theta )+\operatorname {Var} _{G}(Y)=\tau ^{2}+\sigma ^{2}}$, where ${\displaystyle \tau ^{2}}$ is the variance of ${\displaystyle F}$.

Proofedit

let ${\displaystyle F}$ and ${\displaystyle G}$ be probability distributions parameterized with mean a variance asZdroj:https://en.wikipedia.org?pojem=Compound_probability_distribution
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.