In this chapter, we introduce collective risk models. Just as in Chapter 2, we calculate the distribution of the total claim amount, but now we regard the portfolio as a collective that produces a random number where

*N*of claims in a certain time period. We write$$S = X_1 + X_2 + \cdots + X_N$$

(3.1)

*X*_{i}is the*i*th claim. Obviously, the total claims*S*= 0 if*N*= 0. The terms of*S*in (3.1) correspond to actual claims; in (2.26), there are many terms equal to zero, corresponding to the policies that do not produce a claim. We assume that the individual claims*X*_{i}are independent and identically distributed, and also that*N*and all*X*_{i}are independent. In the special case that*N*is Poisson distributed,*S*has a*compound Poisson distribution*. If*N*has a (negative) binomial distribution, then*S*has a*compound (negative) binomial distribution*.## Preview

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