1. Prove that if ?, v are invariant measures, then their convex combination à? + (1-A)v, for ?E( ...

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1. Prove that if ?, v are invariant measures, then their convex combination à? + (1-A)v, for ?E(0, 1) is also an invariant measure. 2. Continuation of Example L2.45. In order to increase the system availability, it is proposed to add a duplicate repair facility so that both computers can be repaired simultaneously. Write down the transition probability matrix, and find availability of this system. 3. Consider the three-state Markov chain with transition matrix [1/5 4/5 01 P = 0 0 1 0 0 Prove that each state is ergodic. Find the mean recurrence times for each state. 4. Consider the Inventory Chain ExampleL2.47 a. Assume the inventory policy a=2, b=3. Find the long run P&L/day for this case. Consider the policy a=1, b=5. Find the transition probability for this inventory chain. Optional: Find the invariant measure, and the long run P&L/day. b.