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Recall that the stochastic process {Xn, n = 0, 1, 2,...} is a discrete-time Markov chain if for each n, Pr(Xn+1 =j | Xn = i, Xn-1 = in-1,..., Xo = io) = Pr(Xn+1 = j | Xn = i) = Pij. Let {Xn, n = 0, 1, 2,...} be a Markov chain, does the following statement hold true as well? Pr(Xn+2 = j| Xn = i, Xn-1 = in-1,..., Xo = io) = Pr(Xn+2 = j | Xn = i). Give a proof if you think it is true, otherwise give a counterexample. Hint: You may use the law of total probability for conditional probability without proof, which was introduced in the proof of Chapman-Kolmogorov Equations.

Recall that the stochastic process {Xn, n = 0, 1, 2,...} is a discrete-time Markov chain if for each n, Pr(Xn+1 =j | Xn = i, Xn-1 = in-1,..., Xo = io) = Pr(Xn+1 = j | Xn = i) = Pij. Let {Xn, n = 0, 1, 2,...} be a Markov chain, does the following statement hold true as well? Pr(Xn+2 = j| Xn = i, Xn-1 = in-1,..., Xo = io) = Pr(Xn+2 = j | Xn = i). Give a proof if you think it is true, otherwise give a counterexample. Hint: You may use the law of total probability for conditional probability without proof, which was introduced in the proof of Chapman-Kolmogorov Equations.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Recall that the stochastic process {Xn, n = 0, 1, 2,...} is a discrete-time Markov chain if for each n, Pr(Xn+1 =j | Xn = i, Xn-1 = in-1,..., Xo = io) = Pr(Xn+1 = j | Xn = i) = Pi,j. Let {Xn, n = 0, 1, 2,...} be a Markov chain, does the following statement hold true as well? Pr(Xn+2 =j | Xn = i, Xn-1 = in-1, ..., Xo = io) = Pr(Xn+2 =j | Xn = i). Give a proof if you think it is true, otherwise give a counterexample. Hint: You may use the law of total probability for conditional probability without proof, which was introduced in the proof of Chapman-Kolmogorov Equations.
Transcribed Image Text:Recall that the stochastic process {Xn, n = 0, 1, 2,...} is a discrete-time Markov chain if for each n, Pr(Xn+1 =j | Xn = i, Xn-1 = in-1,..., Xo = io) = Pr(Xn+1 = j | Xn = i) = Pi,j. Let {Xn, n = 0, 1, 2,...} be a Markov chain, does the following statement hold true as well? Pr(Xn+2 =j | Xn = i, Xn-1 = in-1, ..., Xo = io) = Pr(Xn+2 =j | Xn = i). Give a proof if you think it is true, otherwise give a counterexample. Hint: You may use the law of total probability for conditional probability without proof, which was introduced in the proof of Chapman-Kolmogorov Equations.
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