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Question 1

Suppose that each individual in a large insurance portfolio

incurs losses according to an exponential distribution with mean (1/ λ), where  λ varies over the portfolio according to a G(α, δ) mixing distribution. The respective densities of the two distributions are given by:

(a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(α, δ)

(b) Use the mixing formulation of the Pareto to deduce that if X~P(α, δ), then E(X) = α / (δ – 1)