question on pics

QUESTION 1 (15 marks) Mundo Insurance sold 500 re insurance policies with the following characteristics: Policy Number of Policy Probability of Type Policies Maximum (30 Claim Per Policy (Mt) (as) A 200 400 0.03 B 300 300 0.05 You are given the following information: 0 Claim amounts for each policy1 Xi, are uniformly distributed between 0 and the policy maximum Mi. . The probability of more than one claim per policy is 0. . Claim occurrences are independent. Let S be the total claim amount on the entire portfolio of 500 claims. The insurer has set aside $10 for each of the 500 policies to pay for future claims. Therefore, the insurer has cash reserves with a total amount of $5, 000 available to pay claims. (a) Calculate E(S) and Var(S). (3 marks) (b) Using the normal approximation, calculate the probability that S will exceed $5,000. (1 mark) (c) Calculate the coefficient of skewness of S (find the exact answer, not an approximation). Given your answer for the coefficient of skewness, is the normal approximation a sensible approximation for S for this question? Justify your answer. (2 marks) (d) Using R, write code to simulate 100,000 values of S (ie., simulate 100,000 portfolios, where each portfolio has 200 policies of type A and 300 policies of type B). Plot out the density of the simulated aggregate claims distribution, and use your simulations to estimate the probability that S will exceed $5,000. (3 marks) (e) The insurer sells an additional number of policies of type B, so that the total number of policies of type B is greater than 300. They set aside cash reserves of $15 for each of the additional policies sold. Using the normal approximation, calculate how many additional policies of type B would need to be sold so that the probability that total claims (for all policies) exceed total cash reserves is less than 1%. (6 marks)