1. For a certain collection of contracts, N ∼ Poisson(4), while X ∼ Exp(3). Suppose that each individual claim is subject to a deductible of 2. If S′ is the total amount actually paid on all claims, find the variance of S′.
2. For a certain firm, the number of losses of a certain type has a Poisson(2) distribution. The amount of a loss takes a value of 100, 200 or 300, with probabilities 0.5, 0.3, 0.2, respectively. The firm purchases an insurance policy that will cover all losses above an aggregate deductible of 200. (So, for example, if there was one loss of 100 and three losses of 300, the insurer would pay 800.) What is the expected reimbursement by the insurer?