Question 2) (70 points) A production process has a machine that deteriorates rapidly in both quality and output under heavy usage during the week. Therefore, the machine is inspected at the end of each week. After each inspection, the condition of the machine is noted and classified into one of three possible states: State 1: Good as new, State 2: Operable and State 3: Inoperable. Deterioration is progressive and eventually the machine becomes Inoperable, which may happen any time during a week. If the machine is "Good as new" when it is inspected, it will be "Good as new" at the next inspection epoch with probability (w.p) 0.35, and it will be "Operable" w.p. 0.1. If the machine is "Operable" when it is inspected, it will be "Operable" at the next inspection epoch w.p. 0.3. Once the machine is found in the "Inoperable" condition, which may happen any time during the week, the machine is replaced with a new machine before the next inspection, and its condition becomes "Good as new". a) (10 points) Explain if we can model the above machine deterioration problem as a discrete-time stochastic process. a) (10 points) Explain if we can model the above machine deterioration problem as a discrete-time stochastic process. b) (15 points) Define Xn and write down the P matrix. In particular, what is the probability P(X3=3|X=2,X = 1,X = 1)? c) (15 points) Suppose the machine is just replaced. What is the probability that the machine is in "Operable" condition at the second inspection? d) (15 points) What is the probability that the machine is either in "Good as new" or "Operable" at the end of week three? e) (15 points) What is the probability that the machine is in "Operable" condition for three consecutive inspections before the its condition gets worse