Discussion
Consider the MC consisting of the three states 0, 1, 2 and having TPM :
\[ P = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ 0 & \frac{1}{3} & \frac{2}{3} \end{pmatrix} \]
Which of the following is correct ?
Answer: D
The correct statement is MC is irreducible.
From the matrix, state 0 goes to 1 (\(p_{01} = 1/2\)), 1 goes to 0 and to 2 (\(p_{10} = 1/2\), \(p_{12} = 1/4\)), and 2 goes to 1 (\(p_{21} = 1/3\)). So every state can reach every other through state 1: all states communicate and the chain is irreducible, which rules out A.
A finite irreducible chain has a unique stationary distribution, so B is false. And
\[ \begin{aligned} P_{00}^{(2)} &= \tfrac{1}{2} \cdot \tfrac{1}{2} + \tfrac{1}{2} \cdot \tfrac{1}{2} + 0 \\ &= \tfrac{1}{2} \end{aligned} \]
not 1/4, so C is false.
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