Loss Models: From Data to Decisions, 4th Edition

6.4 The binomial distribution

The binomial distribution is another counting distribution that arises naturally in claim number modeling. It possesses some properties different from the Poisson and the negative binomial that make it particularly useful. First, its variance is smaller than its mean, making it useful for data sets in which the observed sample variance is less than the sample mean. This property contrasts with the negative binomial, where the variance exceeds the mean, and it also contrasts with the Poisson distribution, where the variance is equal to the mean.

Second, it describes a physical situation in which m risks are each subject to claim or loss. We can formalize this situation as follows. Consider m independent and identical risks each with probability q of making a claim.¹ This might apply to a life insurance situation in which all the individuals under consideration are in the same mortality class; that is, they may all be male smokers at age 35 and duration 5 of an insurance policy. In that case, q is the probability that a person with those attributes will die in the next year. Then the number of claims for a single person follows a Bernoulli distribution, a distribution with probability 1 − q at 0 and probability q at 1. The probability generating function of the number of claims per individual is then given by

Now, if there are m such independent individuals, ...