Markov chains

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Problem: The transition matrix of the chain is given, plot the graph with the given probabilities and calculate \( P_{i}(k), i=1,2,3 ; k=1,2 \), assuming that initially the system is in the state of \( S_{1} \). \[ \left(P_{i j}\right)=\left(\begin{array}{cccc} 0.2 & 0.3 & 0.5 & 0 \\ 0.6 & 0 & 0 & 0.4 \\ 0.4 & 0.2 & 0.3 & 0.1 \\ 0.5 & 0 & 0 & 0.5 \end{array}\right) \text {. } \]

15.3.1 Markov chains

2.57 $

Problem: A labeled graph of system states is given. Find: a) the transition matrices in one and two steps, b) the probabilities of the system state after the first, second and the third steps, if at the initial moment the system was in the state of \( S_{2} \), c) final probabilities.

15.3.2 Markov chains

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Problem: Comprise the system of Kolmogorov equations for the probabilities of states of the continuous Markov chain, using the graph of states; find these probabilities, if at the initial moment the system was in the state of \( S_{2} \).

15.3.3 Markov chains

5.14 $

Problem: A labeled graph of a system states is given. Find: a) the transition matrix in one and two steps, b) the probabilities of the system state after the first, second, third steps, if at the initial moment the system was in the state of \( S_{1} \), c) final probabilities.

15.3.4 Markov chains

5.14 $