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If $(X_n)_{n≥0}$ is a Markov chain on $S = {1, 2, 3}$ with initial distribution $α = (1/2, 1/2, 0)$ and transition matrix

$ \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix}, $

then $P(X_2 = 2) = ?$.

My solution:

$X_1 = \begin{bmatrix} 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix}$

$X_2 = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 3/8&3/8&1/4 \end{bmatrix}$

So, $P(X_2=2) = 3/8$


Is this solution correct?

Why or why not?

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  • $\begingroup$ Yes, this is correct. $\endgroup$ – Math1000 May 6 at 20:26
  • $\begingroup$ @Math1000, why is it correct? $\endgroup$ – user366312 May 6 at 20:34

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