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Please solve transition probability matrix (TPM) problems thanks

Questions (12-15): The transition probability matrix (TPM) with four states is defined as follows: \[ P=\begin{array}{r} 0 \\ 1 \\ 2 \end{array}\left[\begin{array}{cccc} 0 & 1 & 2 & 3 \\ 0.16 & 0.2 & 0.25 & 0.39 \\ 0.31 & 0.15 & 0.3 & 0.24 \\ 0.15 & 0.23 & 0.25 & 0.37 \\ 0.21 & 0.35 & 0.15 & 0.29 \end{array}\right] \] 12. The steady-state equations for this Markov Chain, which one is true: \begin{tabular}{|ll} \hline & \( \pi_{0} \) \\ & \( =0.16 \pi_{0} \) \\ & \( +0.31 \pi_{1} \) \\ & \( +0.15 \pi_{2} \) \\ & \( +0.21 \pi_{3} \) \\ \hline \end{tabular} 13. The steady-state probability for state 0 and state 1 are: 1 C A 14. The steady-state probability for state 2 and state 3 are: D A 15. The cost for state 0 is \( \$ 1202 \), state 1 is \( \$ 2105 \), state 2 is \( \$ 3420 \), and state 3 is \( \$ 5210 \) then the total expected cost equals: D

12) The steady-state equations for a Markov chain with a transition probability matrix P and n states can be written as follows: p1 = p1 * P[0][0] + p

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