You are given a transition matrix $P$. Find the steady-state distribution vector. HINT [See Example 4.] $$ P = \begin{bmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix} $$ $\begin{bmatrix} 1/3 & \text{X} & 1/3 & 1/3 \end{bmatrix}$
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Then, we have $\mathbf{v}P = \mathbf{v}$. This gives us the following system of equations: $$ \frac{1}{3}v_1 + \frac{1}{3}v_2 + v_3 = v_1 \\ \frac{1}{3}v_1 + \frac{1}{3}v_2 = v_2 \\ \frac{1}{3}v_1 + \frac{1}{3}v_2 = v_3 \\ v_1 + v_2 + v_3 = 1 $$ Show more…
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