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By Jan Grandell (auth.)

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Suppose we are interested in the number of claims Np in (0, t], which will cost the company more than a certain amount. If the claim costs are independent and if the probability of a claim cost to exceed that amount is p, then Np is a p-thinning. Consider now N and Np as above. Then GNp(s) = E[E[sN~' I Nl] = E [t (~)skpk(1- p)N-k] = E((1- p + ps)N] = GN(1- p(1- s)). Conversely, we say that NP is obtained by p-thinning, and we have GN(s) = GNP(1- 1; 8 ). e. Np is MP(pt, U), and N can be obtained by p-thinning from N 1;p which is MP(tjp, U).

The trivial variable N = 1 is ID but not DID. 55). We will give probabilistic arguments. 27 THE MIXED POISSON DISTRIBUTION If N is DID it is obviously ID, and P{N = 0} > 0 foilows from the representation to be given. Assurne therefore that N is ID and that P{N = 0} > 0. For each n there exist independent and identicaily distributed random variables An,l, ... , An,n so that d N = An,l + An,2 + ... + An,n• Since, as is easily shown, An,l, ... , An,n are non-negative, it foilows that P{N = 0} > 0 implies P{An,k = 0} > 0.

Differentiation Ieads, after simple calculations, to (ß + t- ts)G~(s) = ("'t + at(ß + t)- at 2 s)GN(s). Using 00 00 m=O m=O and equating the coefficients of sm yields, with p_ 1 (t) = 0, the recursion (ß + t) (m + 1) Pm+l (t) = ('y + a(ß + t) + m)tpm(t)- at2Pm-l(t), m Since Po(t) any m. = 0, 1, 2 ... 2 was simply to derive a differential equation for G N, and than to make a MacLaurin expansion. 2 the generating function G N had a simple form, so the differential equation was easily found by derivation.

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