3. A car insurance company is interested in modeling losses from claims coming from a certain class of policy holders for the purposes of pricing and reserving. The policy has a deductible of $500 per claim, up to which the policy holder must pay all costs, and after which the insurance company will pay all additional costs associated with the claim. The insurance company also purchases reinsurance to assist in paying out large claims, which will pay any costs to the insurance company in excess of $15,000. To help with interpreting this policy, some examples of how different-sized claims would be payed out are shown below. Assume the losses for claims on this policy follow an exponential distribution with a mean of 83000 (note: this corresponds with a rate parameter A 1/3000). Example claims: i) A claim carries a loss of $353 dollars. The policy holder must pay the entire $353 associated with the claim, since the cost is less than the deductible for the policy. ii) A claim carries a loss of $2567 dollars. The policy holder pays the deductible of $500, and the insurance company pays the remaining $2067. iii) A claim carries a loss of $32,000 dollars. The policy holder pays the deductible of $500, the insurance company pays $15,000, then reinsurance covers the remaining $16,500. (a) Find the probability that the insurance company needs to pay on a claim they receive (i.e. find the probability that the cost of a claim exceeds the deductible of $500). (3 pts) (b) Given that the insurance company needs to pay on a claim, what is the probability the loss is large enough to be covered by the reinsurance? (i.e., if a loss is at least $500, what is that probability that it is at least $15,500?) (3 pts) (c) Suppose the insurance company has received 1000 claims that have yet to be processed. Assuming that the losses for each claim are independent, find the (exact) sampling distributions of X, the number of these claims the insurance company will need to pay on (claims in excess of 8500), and of Y, the number that will be large enough to require reinsurance (in excess of $15,500). (4 pts) (d) Write an expression for the exact probability that at least 850 claims exceed the policy deductible (i.e. find P(X 850)). (You do not need to compute the value of the sum; simply set it up) (3 pts)