SPLecture5 - Lecture 5 Mariana Olvera-Cravioto Columbia...

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Lecture 5 Mariana Olvera-Cravioto Columbia University [email protected] February 4th, 2015 IEOR 4106, Intro to OR: Stochastic Models Lecture 5 1/18
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Markov chains I Consider a stochastic process { X n : n = 0 , 1 , 2 , . . . } . I X i ∈ S for all i , where S is a countable set. I The elements of S are called the states of the process. I We may assume that S = { 0 , 1 , 2 , 3 , . . . } . I We say that the process is in state i at time n if X n = i . IEOR 4106, Intro to OR: Stochastic Models Lecture 5 2/18
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Markov chains I Consider a stochastic process { X n : n = 0 , 1 , 2 , . . . } . I X i ∈ S for all i , where S is a countable set. I The elements of S are called the states of the process. I We may assume that S = { 0 , 1 , 2 , 3 , . . . } . I We say that the process is in state i at time n if X n = i . IEOR 4106, Intro to OR: Stochastic Models Lecture 5 2/18
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Markov chains I Suppose that whenever the process is in state i , there is a probability P ij of going to state j , i.e., P ( X n +1 = j | X n = i, X n - 1 = i n - 1 , . . . , X 1 = i 1 , X 0 = i 0 ) = P ij for all n 0 . I Note: The above implies that the value of X n +1 only depends on X n , and not on the previous states of the process. I We call such process a discrete time Markov chain . IEOR 4106, Intro to OR: Stochastic Models Lecture 5 3/18
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Transition probabilities I P ij is the probability of transitioning from state i to state j . IEOR 4106, Intro to OR: Stochastic Models Lecture 5 4/18
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Transition probabilities I P ij is the probability of transitioning from state i to state j . I They satisfy: I P ij 0 for all i, j 0 I X j =0 P ij = 1 for all i 0 IEOR 4106, Intro to OR: Stochastic Models Lecture 5 4/18
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Transition probabilities I P ij is the probability of transitioning from state i to state j . I They satisfy: I P ij 0 for all i, j 0 I X j =0 P ij = 1 for all i 0 I The matrix P = P 00 P 01 P 02 · · · P 10 P 11 P 12 · · · . . . . . . . . . P i 0 P i 1 P i 2 · · · . . . . . . . . . is called the one-step transition matrix . IEOR 4106, Intro to OR: Stochastic Models Lecture 5 4/18
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Example: A gambling model I A gambler plays a game where she can either win $1, with probability p , or lose $1, with probability 1 - p . I The gambler starts with a fortune of i 0 and plays the game until she makes $ N or goes bankrupt. I Let X n denote the gambler’s fortune after having played n times; X 0 = i 0 . I Then X n is a Markov chain. IEOR 4106, Intro to OR: Stochastic Models Lecture 5 5/18
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Example: A gambling model I A gambler plays a game where she can either win $1, with probability p , or lose $1, with probability 1 - p .
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