A chain on {0,1,2} has $P=\begin{pmatrix}0&1/2&1/2\\1/2&1/4&1/4\\1/2&1/4&1/4\end{pmatrix}$ and initial probabilities P(X₀=0)=P(X₀=1)=1/4. Find 32E(X₂) (integer).
The initial vector is (1/4,1/4,1/2). After one step it is (3/8,5/16,5/16). After another step it is (5/16,11/32,11/32). Thus E X₂=11/32+2·11/32=33/32 and the requested result is **33**.
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