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Markov Chain

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At the beginning of each year, my car in good, fair, or broken-down condition. A good car will good at the beginning of the next year with the probability 0.85; fair with the probability 0.10; or broken-down quality with a probability of 0.05. A fair car will be fair of the next year with the probability 0.70 or broken-down quality with a probability of 0.30. It costs $6,000 to purchase a good car; a fair car can be traded in for $2,000; a broken-down car has no trade-in value and must immediately be replaced by a good car. It costs $1,000 per year to operate a good car and $1,500 to operate a fair car. Should I replace my car as soon as it becomes fair, or should I drive my car until it breaks-down? Assume that the cost of operating a car during a year depends of the type of car on hand at the beginning of the year (after a new car, if any arrives).

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A Markov chain problem is investgated. The solution will point you in the right direction, but does not give a final solution.

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Markov Chain: Drawing the State Diagram of the Chain

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A particle moves on the states 1, 2, 3, and 4 according to a time-homogeneous Markov chain {Xn ; n ≥ 0 }
with initial distribution p(0) = ( .2, .8, 0, 0 ) and transition matrix : p = [■(.6&.4&0&0@.6&0&.4&0@0&.6&0&.4@0&0&.6&.4)]

a) Draw the state diagram of the chain. Include the transition probabilities in your diagram. (+4)

b) Give P( X1 = 1| X0 = 2) . (+1)

c) Give P( X4 = 2 | X3 = 1) . (+2)

d) Give P( X5 = 4 | X0 = 1, X1 = 2, X2 = 3, X3 = 4, X4 = 3 ) . (+3)

e) Find the unconditional "path" probability P( X0 = 1, X1 = 1, X2 = 2, X3 = 3 ) . (+5)

f) Find p_11^((2)) = P( X2 = 1| X0 = 1) . (+5)

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