mixed Poisson distribution
Notation
Parameters
Support
PMF
Mean
Variance
Skewness
MGF , with the MGF of π
CF
PGF

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]

Definition

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

.

Properties

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[3]

.

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[3]

.

Moment-generating function

The moment-generating function of the mixed Poisson distribution is

.

Examples

Theorem — Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.[3]

Proof

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.

Proof

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

Literature

References

  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.