< Probability distributions < *Compound distribution*

## What is a compound distribution?

A **compound probability distribution** has random variables drawn from a “compound” parametric distribution, where one or more of the distributions parameters (e.g., the mean or variance) are taken from other probability distributions [2]. Compounding distributions make it easier to analyze and visualize data as a while, instead of analyzing separate components.

In simple terms, a compound distribution it isn’t a unique distribution but rather one made up from two or more other probability distributions.

## Formula for compound random variables

Random variables **Y** have a compound distribution if **Y** follows

where

**Y**= a compound random variable,- X
_{j}= a group of iid random variables from one experiment, - N = a non-negative, discrete random variable.

In series form, the formula can be written as the random sum Y = X_{1} + X_{2} + … + X_{N} [3],

where

- The number of terms N is not known (for example, an unknown number of policy claims or customers)
- X
_{i}are iid with common distribution X - Each X
_{i}is independent of N.

## Types of compound distribution

There are many different types of compound probability distributions, each with its own applications. Some of the most common ones include:

*Compound Poisson Distribution:*Used to model situations where there are multiple sources of randomness, such as the number of hits on a website over time. The Poisson distribution models arrivals (i.e. visits to a website) and the compound part just means that the arrival rate is itself a random variable that follows some other underlying distribution.*Compound Gamma Distribution*: Used to model situations where there is an underlying continuous random process with multiple sources of randomness, such as the length of time people spend on your website. The gamma distribution models waiting times (i.e. how long someone spends on your site) and the compound part just means that one of the gamma’s parameters (the shape parameter) is itself a random variable that follows some other underlying distribution.

## Compound vs. mixture distribution

If you’re familiar with chemistry, you may already intuitively know the difference between a compound and a mixture — compound and mixture distributions are the same idea:

**Compound: a substance composed of more than one type of atom bonded together**[4]. In the same way, a compound distribution is “bonded” to make one distribution.**Mixture: a combination of two or more elements or compounds which have not reacted to form a bond**[4]. In the same way, mixture distributions retain features of both parent distributions.

Mixture distributions are formed by merging two or more parent distributions, each representing a different population. An example is blending a normal distribution with a uniform distribution, resulting in a distribution with the mean and variance of the normal distribution and the support of the uniform distribution. On the other hand, compound distributions are formed by conditioning a parent distribution on a categorical variable, which represents a different outcome.

Feature | Mixture distribution | Compound distribution |
---|---|---|

Number of parent distributions | Two or more | One |

Parent distributions | Represent different populations | Represent the same population |

Categorical variable | Not used | Used to condition on a different outcome |

Support | Determined by the parent distributions | Determined by the categorical variable |

*Table of differences between compound and mixture distributions.*

## References

[1] Image retrieved Jan 1 2023 from: Source: http://election.princeton.edu/2014/07/14/senate-control-three-factors-to-watch-in-2014/

[2] A Dictionary of Statistical Terms, 5th edition, prepared for the International Statistical Institute by F.H.C. Marriott. Published for the International Statistical Institute by Longman Scientific and Technical

[3] Applied Probability and Statistics in Actuarial Science and Financial Economics. Retrieved June 29, 2023 from: https://mathmodelsblog.wordpress.com/2010/01/17/an-introduction-to-compound-distributions/