14. The Discrete Hazard Rate Method: Let X denote a nonnegative integer valued random variable. The function ë(ç) = P[X = n\X> n],

ç > 0, is called the discrete hazard rate function.

(a) Show that P{X = ç] = ë(ç) UK ï (1 ~ HO).

(b) Show that we can simulate X by generating random numbers Ui9U2, ... stopping at

(c) Apply this method to simulating a geometric random variable.

Explain, intuitively, why it works.

(d) Suppose that ë(ç) < ñ < 1 for all n. Consider the following algorithm for simulating X and explain why it works: Simulate Xi9 Ui9 i > 1 where Xt is geometric with mean \/p and is a random number.

Set Sk = Xx + · · · + Xk and let