A Markov chain $\{ X_n, n \geqslant 0\}$ with states 0, 1, 2 has the transition probability matrix $$P= \begin{bmatrix} \frac12 & \frac13 & \frac16 \\ 0 & \frac12 & \frac23 \\ \frac12 & 0 & \frac12 \end{bmatrix}$$

If $P\{X_0 =0 \} = P\{X_0=1\}=\frac14$ find $E[X_3]$

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$E[X_3] = 0\cdot P(X_3=0) + 1\cdot P(X_3=1) + 2\cdot P(X_3=2)$ Hence $$E[X_3]= \frac14 P^3_{01} + \frac14P^3_{11} + \frac12P^3_{21} + 2\left[ \frac14P^3_{02} +\frac14P^3_{12} + \frac12P^3_{22}\right]$$


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