Collective risk models

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
$$S = X_1 + X_2 + \cdots + X_N$$
(3.1)
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.

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© Springer-Verlag Berlin Heidelberg 2008

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