5. An insurance company as two-types of auto insurance policies, policy A and policy B. Each...

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5. An insurance company as two-types of auto insurance policies, policy A and policy B. Each policy has claims that follow a Poisson distribution with different means. Policy A has an average of 0.03 claims per year, Policy B has an average of 0.04 claims per year. The payout for each claim also varies per policy: for policy A, it follows an exponential distribution with mean 5,000; for policy B, it follows an exponential distribution with mean 10,000. In the coming year, the insurance company estimates selling 10,000 policies of type A, and 5,000 policies of type B. Let S = SA+SB denote the total claim ammount. (a) (3 points) From a Theorem in class and the textbook you know that the total claim S = SA+ SB is distributed according to a compound Poisson process, i.e. S = X. Determine the probability density function of the severity random variable X. 5. An insurance company as two-types of auto insurance policies, policy A and policy B. Each policy has claims that follow a Poisson distribution with different means. Policy A has an average of 0.03 claims per year, Policy B has an average of 0.04 claims per year. The payout for each claim also varies per policy: for policy A, it follows an exponential distribution with mean 5,000; for policy B, it follows an exponential distribution with mean 10,000. In the coming year, the insurance company estimates selling 10,000 policies of type A, and 5,000 policies of type B. Let S = SA+SB denote the total claim ammount. (a) (3 points) From a Theorem in class and the textbook you know that the total claim S = SA+ SB is distributed according to a compound Poisson process, i.e. S = X. Determine the probability density function of the severity random variable X.