6. (Problem 4.2.4 in Pinsky and Karlin) A component of a computer has an active life, measured in discrete units, that is a random variable S, where k 1 2 3 4 P( = k) 0.4 0.3 0.2 0.1 Suppose that one starts with a fresh component, and each component is replaced by a new component upon failure. Let Xn be the remaining life of the component in service at the end of period n. When Xn = 0, a new item is placed into service at the start of the next period. (a) Set up the transition probability matrix for {Xn}. (b) By showing that the chain is regular and solving for the limiting distribution, determine the long run probability that the item in service at the end of a period has no remaining life and therefore will be replaced. (c) Relate this to the mean life of a component