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STA3007_0506_t05

# STA3007_0506_t05 - STA 3007 Applied Probability Tutorial 5...

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STA 3007 Applied Probability 2005 Tutorial 5 1. The Long Run Behavior of Markov Chains Exercise: i. Suppose that a production process changes state according to a Markov process whose transition probability matrix is given by: P = 0 1 2 3 0 0 . 3 0 . 5 0 . 0 0 . 2 1 0 . 5 0 . 2 0 . 2 0 . 1 2 0 . 2 0 . 3 0 . 4 0 . 1 3 0 . 1 0 . 2 0 . 4 0 . 3 It is known that π 1 = 119 379 = 0 . 3140 and π 2 = 81 379 = 0 . 2137 a Determine the limiting probabilities π 0 and π 3 . b Suppose that states 0 and 1 are ”In-Control” while states 2 and 3 are deemed ”Out of Control”. In the long run, what fraction of time is the process Out of Control? c In the long run, what fraction of transitions are from an In-Control state to an Out- of-Control state? 1

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ii. Suppose that a production process changes state according to a Markov process whose transition probability matrix is given by: P = 0 1 2 3 0 0 . 2 0 . 2 0 . 4 0 . 2 1 0 . 5 0 . 2 0 . 2 0 . 1 2 0 . 2 0 . 3 0 . 4 0 . 1 3 0 . 1 0 . 2 0 . 4 0 . 3 a Determine the limiting distribution for the process.
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STA3007_0506_t05 - STA 3007 Applied Probability Tutorial 5...

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