Discussion
Consider a discrete time Markov chain with state space \(S = \{1, 2, 3\}\) and the transition probability matrix
\[ \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ 0 & \frac{1}{5} & \frac{4}{5} \\ 0 & 1 & 0 \end{bmatrix}. \]
Which one of the following statements is true?
Answer: D
The true statement is 3 is a recurrent state.
Read the matrix row by row. From state 1 the chain can move to 2 or 3, but the first column is zero below the top entry, so no state leads back to 1. From 2 the chain goes to 2 or 3, and from 3 it always goes to 2.
So \(\{2, 3\}\) is a closed communicating class: once there, the chain never leaves, and 2 and 3 reach each other. A finite closed communicating class is recurrent, so both 2 and 3 are recurrent. State 1 is left with probability \(\frac{1}{2}\) at each step and never re-entered, so it is transient.
That rules out the other options: the chain is not irreducible, since 1 cannot be reached from 2 or 3; 1 is transient, not recurrent; and 2 is recurrent, not transient.
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