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Random Processes Markov Chains Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering E-mail: rslab@ntu.edu.tw

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In other words, the probability of any particular future behavior of the process, when its current state is known exactly, is not altered by additional knowledge concerning its past behavior.

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It is customary to label the state space of the Markov chain by the nonnegative integers {0, 1, 2, …} and to use X n =i to represent the process being in state i at time (or stage) n.

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The notation emphasizes that in general the transition probabilities are functions not only of the initial and final states, but also of the time of transition as well.

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The (i+1)st row of P is the probability distribution of the values of X n+1 under the condition that X n =i. If the number of states is finite, then P is a finite square matrix whose order (the number of rows) is equal to the number of states.

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It is also evident that for all time points n, m and all states i 0, …, i n, j 1, …, j m.

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Transition probability matrices of a Markov chain

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Chapman- Kolmogorov equations

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If time is continuous we cannot write down the simultaneous distribution of X(t) for all t. Rather, we pick n, t 1,...,t n and write down probabilities.

If time is continuous we cannot write down the simultaneous distribution of X(t) for all t. Rather, we pick n, t 1,...,t n and write down probabilities.

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