Linear programming problem. Please include all the steps.

2 (5 points). A jewelry store stocks up its l-carat diamond inventory monthly according to this policy: It places an order of 3 diamonds if its 1-carat diamond inventory is empty by the last week of a month; 2 diamonds if it has 1 left; and no order otherwise. The order will be delivered in time for next month's business cycle. It is estimated that the monthly demand Dt for its 1-carat diamonds can be modeled by a Poisson distribution with the rate parameter 2 per month. (a) Construct the transition matrix for the Markov chain. Justify your answer by only including explanations for two entries of the matrix. (b) If X0 = 3, what is the probability that the inventory would be empty in 4 months from now, i.e. P{X4 = Ung = 3}? (c) Find the mean rst-passage time that the store needs to order 3 diamonds to that the store does not need to order any