1. For a certain insurer, the frequency of claims has a negative binomial distribution with an expected value of 16 and the claim severity distribution is Pareto (600, 2). The insurer is planning to introduce a per-claim deductible of 200. If this is done, what would be the reduction in the expected value of aggregate claims?

2. E(X^d)/E(X) is known as the loss elimination ratio (LER), since it gives the proportion of the risk to the insurer that is eliminated by a deductible of d. For each of the following distributions, find the LER in terms of d and the parameters. (a) X ∼ Exp(λ), (b) X ∼ Pareto(Θ, λ), (c) X ∼ Gamma(2, λ;). In each case verify that your answers have the correct limits as d approaches 0 or ∞.