Section 1

# Section 1 - Notes for Section Feb 1-5 ORIE 3510 Problem 4.5...

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Notes for Section Feb 1-5 ORIE 3510 Problem 4.5 Have Markov chain { X n , n = 0 , 1 ,... } with transition matrix P = 1 / 2 1 / 3 1 / 6 0 1 / 3 2 / 3 1 / 2 0 1 / 2 and initial distribution α i = P ( X 0 = i ) = 1 / 4 i = 0 1 / 4 i = 1 1 / 2 i = 2 . Get 3-step transition matrix P (3) = P 3 = 13 / 36 11 / 54 47 / 108 4 / 9 4 / 27 11 / 27 5 / 12 2 / 9 13 / 36 1

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and use to compute unconditional expected value of X 3 as EX 3 = P ( X 3 = 1) + 2 P ( X 3 = 2) = 2 X i =0 P ( X 3 = 1 | X 0 = i ) P ( X 0 = i ) + 2 2 X i =0 P ( X 3 = 2 | X 0 = i ) P ( X 0 = i ) = 2 X i =0 p (3) i 1 α i + 2 2 X i =0 p (3) i 2 α i = 1 4 ± 11 54 + 2 47 108 ² + 1 4 ± 4 27 + 2 11 27 ² + 1 2 ± 2 9 + 2 13 36 ² = 0 . 981 . Problems 4.10, 4.11 From Example 4.3: model Gary’s mood as a Markov chain { X n } on state- space { 0 = cheerful , 1 = so-so , 2 = glum } with transition matrix P = 0 . 5 0 . 4 0 . 1 0 . 3 0 . 4 0 . 3 0 . 2 0 . 3 0 . 5 . We are interested in computing the probability P ( X 1 6 = 2 ,X 2 6 = 2 ,X 3 6 = 2 | X 0 = 0). Consider instead an auxiliary Markov chain Y n with transition matrix e P = 0 . 5 0 . 4 0 . 1 0 . 3 0 . 4 0 . 3 0 0 1 . Then, from states 0 or 1,
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## This note was uploaded on 10/29/2011 for the course ORIE 3510 taught by Professor Resnik during the Spring '09 term at Cornell.

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Section 1 - Notes for Section Feb 1-5 ORIE 3510 Problem 4.5...

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