negative binomial distribution

The negative binomial distribution is the count distribution actuaries turn to when claim numbers are more scattered than a Poisson can capture. The Poisson insists the variance equals the mean, but real-world claim counts often jump around far more — some policyholders are simply riskier than others. The negative binomial gracefully allows the variance to exceed the mean, making it the standard model for overdispersed counts.

It can be reached two ways, which is part of its charm. Classically it counts the number of failures before the r-th success in repeated independent trials. But the version actuaries love comes from mixing: take a Poisson whose rate λ is itself random, varying from policyholder to policyholder according to a gamma distribution, and the resulting count is negative binomial. This 'Poisson-gamma mixture' captures the idea that the population is a blend of high-risk and low-risk individuals. Its mean is rβ and its variance is rβ(1+β), always larger than the mean since β > 0.

In general insurance the negative binomial is often the better frequency model precisely because it admits heterogeneity: not every driver, household, or business carries the same underlying claim rate. It is a member of the (a,b,0) class used with the Panjer recursion to build aggregate loss distributions, and its extra flexibility usually produces more realistic — and more prudent — estimates of the chance of a heavy claim year than a Poisson would.