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Linear AlgebraMediumMarkov Chains & Long-Run Behavior

Markov Chains & Long-Run Behavior · Medium problem

A chain on states 1, 2, 3 has transition matrix M=(1/21/401/21/21/201/41/2)M = \begin{pmatrix} 1/2 & 1/4 & 0 \\ 1/2 & 1/2 & 1/2 \\ 0 & 1/4 & 1/2 \end{pmatrix}, where MijM_{ij} is the probability of moving to state ii from state jj. Starting in state 1, what is the expected number of steps until the chain first returns to state 1?