Practice Questions 4 – ACTSC 431/831, FALL 20111. Prove that ifXhas a mixed Poisson distribution and the mixing distribution is infinitelydivisible, thenXis infinitely divisible.2. LetNLbe the number of losses. The size of thejth loss isXj. Assume thatNL, X1, X2, ...areindependent andX1, X2, ...have the same distribution as a Pareto distributionPareto(3,100).A loss will result in a payment when the loss exceeds an ordinary deductible of 50. LetNPbethe number of payments andNP*be the number of losses which will not result in payments.(a) Assume thatNLis infinitely divisible. Show thatNP*is infinitely divisible.(b) Assume thatNLhas the same distribution as that the compound frequency model∑Ni=1Mi, where the primary distribution is a negative binomialNB(4,9) and the seconddistribution is a Poisson distribution with mean 8.i. Calculate the variance of the number of payments.ii. CalculateCov(NP, NP*).
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Probability theory, Cumulative distribution function, Discrete probability distribution