Consider a machine whose condition at any time can be observed and classified as being in one of the following three states: State 1: Good operating order State 2: Deteriorated operating order State 3: In repair We observe the condition of the machine at the end of each period in a sequence of periods. Let X_n denote the condition of the machine at the end of period n for n = 1, 2, ... Let X_0 be the condition of the machine at the start. We assume that the sequence of machine conditions is a Markov chain with transition probabilities P_11 = 0.9, P_12 = 0.1, P_13 = 0, P_21 = 0, P_22 = 0.9, P_23 = 0.1, P_31 = 1, P_32 = 0, P_33 = 0, and that the process starts in state X_0 = 1. 1. Find Pr{X_4 = 1}.2. Calculate the limiting distribution. 3. What is the long run rate of repairs per unit time?