The Discrete Hazard Rate Method: Let X denote a nonnegative integer valued random variable. The…

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

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(a) Show that P{X = n} = λ(n) width=

(b) Show that we can simulate X by generating random numbers U1,U2, . . . stopping at

X = min{n: Un  width=  λ(n)}

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

intuitively, why it works.

(d) Suppose that λ(n)  width=p <1 for all n. Consider the following algorithm for simulating X and explain why it works: Simulate Xi,Ui , i  width= 1 where Xi is geometric with mean 1/p and Ui is a random number. Set Sk = X1 +···+Xk

and let

X = min {Sk: Uk  width= λ(Sk)/p}