Loss Models: From Data to Decisions, 4th Edition

6.1 Introduction

The purpose of this chapter is to introduce a large class of counting distributions. Counting distributions are discrete distributions with probabilities only on the nonnegative integers; that is, probabilities are defined only at the points 0, 1, 2, 3, 4,…. In an insurance context, counting distributions can be used to describe the number of events such as losses to the insured or claims to the insurance company. An understanding of both the number of claims and the size of claims provides a deeper understanding of a variety of issues surrounding insurance payments than if information is only available about total losses. The description of total losses in terms of numbers and amounts separately makes it possible to address issues of modification of an insurance contract. Another reason for separating numbers and amounts of claims is that models for the number of claims are fairly easy to obtain and experience has shown that the commonly used distributions really do model the propensity to generate losses.

We now formalize some of the notation used for models for discrete phenomena. The probability function (pf) p_k denotes the probability that exactly k events (such as claims or losses) occur. Let N be a random variable representing the number of such events. Then

As a reminder, the probability generating function (pgf) of a discrete random variable N with pf ...