If X is a positive integer valued random variable, with mass function pi =

P(X = i), i ≥ 1, then the function

λ(i) = P(X = i|X ≥ i)

is called the (discrete) hazard rate function of X.

(a) Express P(X >n) in terms of the values λ(i), i ≥ 1.

(b) If λ(i) is increasing (decreasing) in i then the random variable X is said to have increasing (decreasing) failure rate. Let X

∗

n be a random variable whose distribution is that of the conditional distribution of X − n given that X ≥ n. That is, P(X

∗

n

= j) = P(X = n +j |X ≥ n).

Show that if X has increasing (decreasing) failure rate it and only if X

∗

n stochastically decreases (increases) in n.