Answer the following parts separately:
(a) Suppose that the initial surplus is 10. How large should the adjustment coefficient R be, to ensure that the probability of ruin will be less than 0.10?
(b) Suppose that the adjustment coefficient R = 2/9, λ = 5, and X is exponentially distributed with mean 3. Find c.
(c) You are given that c = 20, λ = 5. All you know about X is that E(X) = 3. What is the probability that the surplus will eventually drop below its initial value?
(d) Suppose that X takes some fixed value with certainty. Show that if the initial surplus does drop below its initial value, the amount of deficit the first time this occurs will be uniformly distributed.