Claim count distribution ($p_n=\mathbb{P}(N=n)$)

The probability distribution of the nonnegative integer number of claims over a specified accounting period. Its probability generating function determines the count probabilities. A Poisson distribution gives equal count expected value and variance; a Poisson mixture allows heterogeneity in the underlying risk intensity.

The nonnegative integer laws whose probabilities satisfy this recurrence for all $n\ge 1$. Their probability generating function solves $(1-az)G^{\prime }(z)=(a+b)G(z)$. This is the $(a,b,0)$ count class used in the Panjer aggregate algorithm; the count recurrence itself should be distinguished from that aggregate recursion. It includes the Poisson distribution, binomial distribution, and negative binomial distribution with suitable parameters.