A compound probability distribution has random variables drawn from a “compound” parametric distribution, where one or more of the distributions parameters (i.e. the mean) are taken from other probability distributions. In other words, it is a probability distribution that is based on two or more other probability distributions.

Compound probability distributions are used in statistics and probability theory to model situations where there is more than one source of randomness. For example, if you were trying to model the number of hits on a website over time, you might use a compound Poisson distribution, where the arrival rate of new visitors (i.e. the parameter that governs how often new visitors come to the site) is itself a random variable that follows some other distribution.

Another example would be if you were trying to model the length of time people spend on your website. Here you might use a gamma distribution for the length of time each person spends on the site, where the shape parameter of the gamma distribution is itself a random variable that follows some other distribution.

## 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.

## Conclusion

In conclusion, a compound probability distribution is a type of probability distribution that has random variables drawn from a “compound” parametric distribution, where one or more of the distributions parameters (i.e. the mean) are taken from other probability distributions. Compound probability distributions are used in statistics and probability theory to model situations where there is more than one source of randomness.