4. (25 points) Emily's bicycie store sells a particular model that can be ordered weekly from a supplier. Weekly demands are independent and identically distributed random variables. Let 13,. represent the demand in the nth week and X" be the number of bicycles on hand at the beginning ofweelc 1:. Assume that X.) = 2 and 13(1),. = 0) = %,P(Dn = 1) = P(D,. = 2) = , and accessive demands are lost. At the beginning of week 11, the store plaoa an order that is delivered immediately. The store follows the following ordering policy: if the inventory at the beginning of a week is less than or equal to 1' order 1 bicycle; if the inventory at the beginning of week is 2, do not order. (a) (6 points) Fbrmulate the inventory problem as a. DTMC and calculate the one-step tran ltion probability matrix. (b) (6 points) What is the probability that there is no bicycle on hand at the beginning of week 3'? (c) (6 points) What is the expected number of lost sales (unsatised excessive demand) per week in the long run? (d) (7 points) Suppose that it costs $50 to hold a bicycle for a week and the net income (selling price - purchasing cost) for each bicycle sold is $1M. What is the expected prot per week in the long run