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

2. 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.

3. 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.

4. 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.

7. 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.

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

17. Transition probability matrices of a Markov chain

19. Chapman- Kolmogorov equations