7. (16 marks) Consider a machine whose condition at any time can be classified as being in one of the following three states: State 1: Good operating order State 2: Deteriorated operating order State 3: In repair The condition of the machine is observed at the end of each period in a sequence of periods. Let X, denote the condition of the machine at the end of period n for n = 1, 2, .... Let Xo be the condition of the machine at the start. The sequence of machine conditions X, is a Markov chain with transition probabilities PIl = 0.9 P12 = 0.1 P13 = 0 P21 = 0 P22 = 0.9 Pas = 0.1 P31 = 1 P32 = 0 P33 = 0 and the process starts in state Xo = 1. a) Find P(X4 = 1). b) Calculate the limiting distribution. c) What is the long run rate of repairs per unit time