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.

[1] It should not be confused with compound Poisson distribution or compound Poisson process.

[2] A random variable X satisfies the mixed Poisson distribution with density π(λ) if it has the probability distribution[3] If we denote the probabilities of the Poisson distribution by qλ(k), then In the following let

π ( λ )

be the expected value of the density

π ( λ )

π ( λ )

be the variance of the density.

The expected value of the mixed Poisson distribution is For the variance one gets[3] The skewness can be represented as The characteristic function has the form Where

is the moment generating function of the density.

For the probability generating function, one obtains[3] The moment-generating function of the mixed Poisson distribution is Theorem — Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.

π ( λ ) =

distributed random variable.

Theorem — Compounding a Poisson distribution with rate parameter distributed according to an exponential distribution yields a geometric distribution.

π ( λ ) =

1 β

λ β

1 β

distributed random variable.

Using integration by parts n times yields:

1 β

λ β

β

1 + β

β

β

β