# Expected value of Markov chain after nth steps

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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• Ask on Mathematics instead. Jul 29, 2022 at 10:06
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– D.W.
Jul 29, 2022 at 17:58

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