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Consider a Markov Chain with state space $\{0,1,2,3,4\}$ and transition matrix

$$P=\begin{matrix} & \begin{matrix}0&&1&&2&&3 && 4\end{matrix} \\\\ \begin{matrix}0\\\\1\\\\2\\\\3\\\\4\end{matrix} & \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\\\  1 / 3 & 1 / 3 & 1 / 3 & 0 & 0 \\\\  0 & 1 / 3 & 1 / 3 & 1 / 3 & 0 \\\\ 0 & 0 & 1 / 3 & 1 / 3 & 1 / 3 \\\\  0 & 0 & 0 & 0 & 1\end{pmatrix}\\\\ \end{matrix}$$

Then $\lim _{n \rightarrow \infty} p_{23}^{(n)}$ equals

1. $\frac{1}{3}$

2. $\frac{1}{2}$

3. 0

4. 1
in Probability by Expert (2.4k points) | 90 views

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