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 N of claims in a certain time period. We write
where Xi is the ith 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 Xi are independent and identically distributed, and also that N and all Xi 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.
$$S = X_1 + X_2 + \cdots + X_N$$
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