7. If X is a positive integer valued random variable, with mass function , , then the function is called the (discrete) hazard rate function of X.

(a) Express in terms of the values .

(b) If is increasing (decreasing) in i then the random variable X is said to have increasing (decreasing) failure rate. Let be a random variable whose distribution is that of the conditional distribution of given that . That is, Show that if X has increasing (decreasing) failure rate it and only if stochastically decreases (increases) in n.