lec0201 - IEOR 4106: Professor Whitt Lecture Notes,...

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Unformatted text preview: IEOR 4106: Professor Whitt Lecture Notes, Tuesday, February 1 More on Markov Chains 1. Structure of Markov Chain Transition Matrices Which of the following is a Markov chain transition matrix? And why?: (a) P = . 1 0 . 0 0 . 0 0 . 9 0 . . 0 0 . 4 0 . 5 0 . 0 0 . 6 . 3 0 . 3 0 . 0 0 . 4 0 . . 3 0 . 0 0 . 0 0 . 7 0 . . 0 0 . 7 0 . 0 0 . 0 0 . 3 (b) P = . 1 . . 0 0 . 9 0 . . . 4 . 0 0 . 0 0 . 6 . 6- . 3 0 . 0 0 . 4 0 . . 3 . . 0 0 . 7 0 . . . 7 . 0 0 . 0 0 . 3 (c) P = . 1 0 . 0 0 . 0 0 . 9 0 . . 0 0 . 4 0 . 0 0 . 0 0 . 6 . 3 0 . 3 0 . 0 0 . 4 0 . . 3 0 . 0 0 . 0 0 . 7 0 . . 0 0 . 7 0 . 0 0 . 0 0 . 3 - What is special about the following Markov chain transition matrices?: (d) P = . 1 0 . 9 0 . 0 0 . 0 0 . . 6 0 . 4 0 . 0 0 . 0 0 . . 0 0 . 0 0 . 3 0 . 4 0 . 3 . 0 0 . 0 0 . 3 0 . 7 0 . . 0 0 . 0 0 . 0 0 . 7 0 . 3 (e) P = 1 . 0 0 . 0 0 . 0 0 . 0 0 . . 1 0 . 4 0 . 0 0 . 0 0 . 5 . 3 0 . 3 0 . 0 0 . 4 0 . . 3 0 . 0 0 . 0 0 . 7 0 . . 0 0 . 1 0 . 2 0 . 4 0 . 3 (c) P = . 1 0 . 0 0 . 0 0 . 9 0 . . 0 0 . 4 0 . 0 0 . 0 0 . 6 . 3 0 . 3 0 . 0 0 . 4 0 . . 3 0 . 0 0 . 0 0 . 7 0 . . 0 0 . 7 0 . 0 0 . 0 0 . 3 2. Classification of States See Section 4.3. Concepts: 1. State j is accessible from state i if it is possible to get to j from i in some finite number of steps. (notation: i j ) It does not have to be in a single step; it can be in several steps. In other words, i j if P ( n ) i,j > 0 for some n , i.e., if the n-step transition probability from state i to state j is strictly positive....
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lec0201 - IEOR 4106: Professor Whitt Lecture Notes,...

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