The Delaporte distribution, also called the Poisson*negative binomial distribution, has gained popularity in actuarial science as a way to model certain events, such as the expected number of insurance claims during a time period. This discrete probability distribution can be defined as a special cases of a mixed Poisson or as a convolution of negative binomial and Poisson distributions; it has less variability than the negative binomial but more variability than the Poisson. Alternatively, it can be thought of as a counting distribution with negative binomial and Poisson components.
The three-parameter Delaporte distribution has been put forward as an alternative to the two-parameter gamma mixture (aka the negative binomial distribution) .
Delaporte distribution properties
The Delaporte distribution has a fixed and variable component to its mean parameter, similar to how the negative binomial can be seen as a Poisson distribution with a gamma distributed random variable for a mean. Think of the Delaporte distribution as a compound Poisson distribution, where the mean parameter has two components:
- A fixed Poisson component (λ), which must be strictly positive,
- A gamma distributed component (with parameters α, β). Both parameters must be strictly positive .
Other properties :
- α, β λ
- Mean = λ + αβ
- Mode = z, z + 1, if z is an integer, where z = (α – 1)/β + λ; [z] elsewhere.
- Variance: λ + α β (β + 1).
Special cases of the Delaporte distribution
Special cases of the Delaporte distribution :
- Delaporte (λ, a, 0) = Poisson distribution (λ),
- Delaporte (0, α, b) = Polya distribution (α, β),
- Delaporte(0, 1, b) = Geometric distribution (1/(1 + β)).
This distribution was first analyzed by Pierre Delaporte  in relation to automobile accident claim counts back in 1959. However, its roots go back even further to 1934 when it made an appearance in a paper by Rolf von Lüders  under the moniker “The Formel II Distribution.”
 Norman L. Johnson, Adrienne W. Kemp, Samuel Kotz. (2005) Univariate discrete distributions. Wiley.
 Avraham, CC BY-SA 3.0 US https://creativecommons.org/licenses/by-sa/3.0/us/deed.en, via Wikimedia Commons
 Adler, A. (2013). Delaporte: Statistical Functions for the Delaporte Distribution. Retrieved April 11, 2013 from: https://www.researchgate.net/publication/316636105_Delaporte_Statistical_Functions_for_the_Delaporte_Distribution
 Vose, D. Risk Analysis: A Quantitative Guide
 Delaporte, Pierre J. (1960). “Quelques problèmes de statistiques mathématiques poses par l’Assurance Automobile et le Bonus pour non sinistre” [Some problems of mathematical statistics as related to automobile insurance and no-claims bonus]. Bulletin Trimestriel de l’Institut des Actuaires Français (in French). 227: 87–102