mixed Poisson distribution
MGF , with the MGF of π

A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that a random variable is Poisson distributed, where 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 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



For the variance one gets[3]



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[3]


Moment-generating function

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.[3]


Let be a density of a distributed random variable.

Therefore we get .

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


Let be a density of a distributed random variable. Using integration by parts n times yields:

Therefore we get .

Table of mixed Poisson distributions

mixing distribution mixed Poisson distribution[4]
gamma negative binomial
exponential geometric
inverse Gaussian Sichel
Poisson Neyman
generalized inverse Gaussian Poisson-generalized inverse Gaussian
generalized gamma Poisson-generalized gamma
generalized Pareto Poisson-generalized Pareto
inverse-gamma Poisson-inverse gamma
log-normal Poisson-log-normal
Lomax Poisson–Lomax
Pareto Poisson–Pareto
Pearson’s family of distributions Poisson–Pearson family
truncated normal Poisson-truncated normal
uniform Poisson-uniform
shifted gamma Delaporte
beta with specific parameter values Yule



  1. ^ Willmot, Gordon E.; Lin, X. Sheldon (2001), "Mixed Poisson distributions", Lundberg Approximations for Compound Distributions with Insurance Applications, New York, NY: Springer New York, vol. 156, pp. 37–49, doi:10.1007/978-1-4613-0111-0_3, ISBN 978-0-387-95135-5, retrieved 2022-07-08
  2. ^ Willmot, Gord (1986). "Mixed Compound Poisson Distributions". ASTIN Bulletin. 16 (S1): S59–S79. doi:10.1017/S051503610001165X. ISSN 0515-0361.
  3. ^ a b c d Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Cambridge. pp. 5–7. doi:10.1017/S051503610001165X.((cite web)): CS1 maint: url-status (link)
  4. ^ Karlis, Dimitris; Xekalaki, Evdokia (2005). "Mixed Poisson Distributions". International Statistical Review / Revue Internationale de Statistique. 73 (1): 35–58. ISSN 0306-7734.