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Practice Questions 4

Practice Questions 4 - Practice Questions 4 ACTSC 431/831...

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Practice Questions 4 – ACTSC 431/831, FALL 2011 1. Prove that if X has a mixed Poisson distribution and the mixing distribution is infinitely divisible, then X is infinitely divisible. 2. Let N L be the number of losses. The size of the j th loss is X j . Assume that N L , X 1 , X 2 , ... are independent and X 1 , X 2 , ... have the same distribution as a Pareto distribution Pareto (3 , 100). A loss will result in a payment when the loss exceeds an ordinary deductible of 50. Let N P be the number of payments and N P * be the number of losses which will not result in payments. (a) Assume that N L is infinitely divisible. Show that N P * is infinitely divisible. (b) Assume that N L has the same distribution as that the compound frequency model N i =1 M i , where the primary distribution is a negative binomial NB (4 , 9) and the second distribution is a Poisson distribution with mean 8. i. Calculate the variance of the number of payments. ii. Calculate Cov ( N P , N P * ).
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