14. The Discrete Hazard Rate Method: Let X denote a nonnegative integer valued random variable. The function λ(n) = P{X = n | X n}, n 0, is called the discrete hazard rate function.

(a) Show that P{X = n} = λ(n)

2n−1 i=0 (1 − λ(i)).

(b) Show that we can simulate X by generating random numbers U1,U2, . . . stopping at X = min{n: Un λ(n)}

(c) Apply this method to simulating a geometric random variable. Explain, intuitively, why it works.

(d) Suppose that λ(n)p 1 where Xi is geometric with mean 1/p and Ui is a random number. Set Sk = X1 + · · · + Xk and let

Transcribed Image Text:

X= min{S: Uk