Let V be a non-negative random variable with density fv and moment generating function Mv . You can assume that the moment generating function exists on the whole real line.

(a) As a preliminary, let X be an exponential () random variable. For x > 0, provide the formula for P(X > x). You don't have to prove it.

(b) For t > 0, explain why 0 < MV (t) < 1. You cannot appeal to Part (c) as the reason.

(c) For all t > 0, find an event At such that MV (t) = P(At). The event At should involve V and an exponential random variable that is independent of V and has a rate that involves t. Suggestion: Write MV (t) as an integral and apply Part (a) appropriately