1. Show that an exponential random variable such that the inverse of the parameter is gamma-distributed is Pareto-distributed. More precisely, show that if X | M = m ∈ Exp(m) with M-1 ∈ Γ(p, a), then X has a (translated) Pareto distribution.
2. Let X and Y be random variables such that Y | X = x ∈ Exp(1/x) with X ∈ Γ(2, 1).
(a) Show that Y has a translated Pareto distribution.
(b) Compute E Y .
(c) Check the value in (b) by recomputing it via our favorite formula for conditional means.