Mixed Poisson distribution

A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution, and that the rate parameter itself is considered as a random variable. Hence it is a special case of a compound probability distribution. Mixed Poisson distributions can be found in actuarial mathematics as a general approach for the distribution of the number of claims and is also examined as an epidemiological model. It should not be confused with compound Poisson distribution or compound Poisson process.

Definition

A random variable X satisfies the mixed Poisson distribution with density (λ) if it has the probability distribution

If we denote the probabilities of the Poisson distribution by , then

Properties

• The variance is always bigger than the expected value. This property is called overdispersion. This is in contrast to the Poisson distribution where mean and variance are the same.
• In practice, almost only densities of gamma distributions, logarithmic normal distributions and inverse Gaussian distributions are used as densities (λ). If we choose the density of the gamma distribution, we get the negative binomial distribution, which explains why this is also called the Poisson gamma distribution.

In the following let be the expected value of the density and be the variance of the density.

Expected value

The expected value of the mixed Poisson distribution is

Variance

For the variance one gets

Skewness

The skewness can be represented as

Characteristic function

The characteristic function has the form

Where is the moment generating function of the density.

Probability generating function

For the probability generating function, one obtains

Moment-generating function

The moment-generating function of the mixed Poisson distribution is

Examples

Table of mixed Poisson distributions

References

Further reading