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:
- 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