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  parameter, and a gamma-distributed variable component, which has the  and  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 Lüders, where it was called the Formel II distribution. Properties
The skewness of the Delaporte distribution is:

The excess kurtosis of the distribution is:

References

Further reading

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Discrete distributions
Compound probability distributions