Let the random sum be the aggregate claims generated in a fixed period by an independent group of insureds. When the number of claims follows a Poisson distribution, the sum is said to have a compound Poisson distribution. When the number of claims has a mixed Poisson distribution, the sum is said to have a compound mixed Poisson distribution. A mixed Poisson distribution is a Poisson random variable such that the Poisson parameter is uncertain. In other words, is a mixture of a family of Poisson distributions and the random variable specifies the mixing weights. In this post, we present several basic properties of compound mixed Poisson distributions. In a previous post (Compound negative binomial distribution), we showed that the compound negative binomial distribution is an example of a compound mixed Poisson distribution (with gamma mixing weights).

In terms of notation, we have:

- ,
- Poisson,
- some unspecified distribution.

The following presents basic proeprties of the compound mixed Poisson in terms of the mixing weights and the claim amount random variable .

**Mean and Variance**

**Moment Generating Function**

**Cumulant Generating Function**

**Measure of Skewness**

Measure of skewness:

**Previous Posts on Compound Distributions**

An introduction to compound distributions

Some examples of compound distributions

Compound Poisson distribution

Compound Poisson distribution-discrete example

Compound negative binomial distribution

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