6. A new item will fail on its ith day of use with probability

. An item that fails during a period is replaced by a new one at the beginning of the next period. Let denote the age of the item in use at the beginning of period n. That is, if the item in use is beginning its ith day. The random variables can be interpreted as the age at time n of a renewal process whose interarrival times have mass function , with signifying that a renewal occurs at time n.

(a) Argue that is a Markov chain and give its transition probabilities.

(b) Suppose . If decreases in i, show that stochastically increases in n.