# Delaporte distribution

Parameters Probability mass function When ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are 0, the distribution is the Poisson.When ${\displaystyle \lambda }$ is 0, the distribution is the negative binomial. Cumulative distribution function When ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are 0, the distribution is the Poisson.When ${\displaystyle \lambda }$ is 0, the distribution is the negative binomial. ${\displaystyle \lambda >0}$ (fixed mean) ${\displaystyle \alpha ,\beta >0}$ (parameters of variable mean) ${\displaystyle k\in \{0,1,2,\ldots \}}$ ${\displaystyle \sum _{i=0}^{k}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{k-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(k-i)!}}}$ ${\displaystyle \sum _{j=0}^{k}\sum _{i=0}^{j}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{j-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(j-i)!}}}$ ${\displaystyle \lambda +\alpha \beta }$ ${\displaystyle {\begin{cases}z,z+1&\{z\in \mathbb {Z} \}:\;z=(\alpha -1)\beta +\lambda \\\lfloor z\rfloor &{\textrm {otherwise}}\end{cases}}}$ ${\displaystyle \lambda +\alpha \beta (1+\beta )}$ See #Properties See #Properties ${\displaystyle {\frac {e^{\lambda (e^{t}-1)}}{(1-\beta (e^{t}-1))^{\alpha }}}}$

The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science.[1][2] It can be defined using the convolution of a negative binomial distribution with a Poisson distribution.[2] Just as the negative binomial distribution can be viewed as a Poisson distribution where the mean parameter is itself a random variable with a gamma distribution, the Delaporte distribution can be viewed as a compound distribution based on a Poisson distribution, where there are two components to the mean parameter: a fixed component, which has the ${\displaystyle \lambda }$ parameter, and a gamma-distributed variable component, which has the ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ parameters.[3] The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959,[4] although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders,[5] where it was called the Formel II distribution.[2]

## Properties

The skewness of the Delaporte distribution is:

${\displaystyle {\frac {\lambda +\alpha \beta (1+3\beta +2\beta ^{2})}{\left(\lambda +\alpha \beta (1+\beta )\right)^{\frac {3}{2}}}}}$

The excess kurtosis of the distribution is:

${\displaystyle {\frac {\lambda +3\lambda ^{2}+\alpha \beta (1+6\lambda +6\lambda \beta +7\beta +12\beta ^{2}+6\beta ^{3}+3\alpha \beta +6\alpha \beta ^{2}+3\alpha \beta ^{3})}{\left(\lambda +\alpha \beta (1+\beta )\right)^{2}}}}$

## References

1. ^ Panjer, Harry H. (2006). "Discrete Parametric Distributions". In Teugels, Jozef L.; Sundt, Bjørn (eds.). Encyclopedia of Actuarial Science. John Wiley & Sons. doi:10.1002/9780470012505.tad027. ISBN 978-0-470-01250-5.
2. ^ a b c Johnson, Norman Lloyd; Kemp, Adrienne W.; Kotz, Samuel (2005). Univariate discrete distributions (Third ed.). John Wiley & Sons. pp. 241–242. ISBN 978-0-471-27246-5.
3. ^ Vose, David (2008). Risk analysis: a quantitative guide (Third, illustrated ed.). John Wiley & Sons. pp. 618–619. ISBN 978-0-470-51284-5. LCCN 2007041696.
4. ^ 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.
5. ^ von Lüders, Rolf (1934). "Die Statistik der seltenen Ereignisse" [The statistics of rare events]. Biometrika (in German). 26: 108–128. doi:10.1093/biomet/26.1-2.108. JSTOR 2332055.