# ⓘ Delaporte distribution. The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined us ..

## ⓘ Delaporte distribution

The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a negative binomial distribution with a Poisson distribution. 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 λ {\displaystyle \lambda } parameter, and a gamma-distributed variable component, which has the α {\displaystyle \alpha } and β {\displaystyle \beta } parameters. The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959, although it appeared in a different form as early as 1934 in a paper by Rolf von Luders, where it was called the Formel II distribution.

## 1. Properties

The skewness of the Delaporte distribution is:

λ + α β 1 + 3 β + 2 β 2 λ + α β 1 + β) 3 2 {\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:

λ + 3 λ 2 + α β 1 + 6 λ + 6 λ β + 7 β + 12 β 2 + 6 β 3 + 3 α β + 6 α β 2 + 3 α β 3 λ + α β 1 + β) 2 {\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}}}}