# Given Information A Markov chain has the following transition matrix: \[ P = \begin{pmatrix} & 1 & \\ \frac{1}{2} & & \frac{1}{2} \\ & 1 & \end{pmatrix} \] # Definitions - **Irreducible:** A Markov chain is irreducible if it is possible to get from any state to any other state (perhaps in multiple steps). - **Stationary Distribution:** A probability vector \(\pi\) such that \(\pi P = \pi\) and \(\sum_i \pi_i = 1\). - **Ergodic:** A chain is ergodic if it is irreducible and aperiodic (not trapped in cycles). --- ## 1. Is the chain irreducible? Check if every state can be reached from every other state in some number of steps: - From state 1: can go to state 2 in 1 step. - From state 2: can go to state 1 or 3 in 1 step. - From state 3: can go to state 2 in 1 step. Check paths: - From 1 → 2 → 3: State 1 can reach 3 in 2 steps. - From 3 → 2 → 1: State 3 can reach 1 in 2 steps. - All states can reach each other. **Conclusion:** The chain is irreducible. --- ## 2. Find the stationary distribution, if it exists Let \(\pi = (\pi_1, \pi_2, \pi_3)\) such that \(\pi P = \pi\): \[ \begin{align*} \pi_1 &= \pi_1 + \frac{1}{2} \pi_2 + \pi_3 = \frac{1}{2} \pi_2 \\ \pi_2 &= 1\pi_1 + \pi_2 + 1\pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= \pi_1 + \frac{1}{2} \pi_2 + \pi_3 = \frac{1}{2} \pi_2 \\ \end{align*} \] Also, \(\pi_1 + \pi_2 + \pi_3 = 1\). From above, \(\pi_1 = \pi_3\). Let \(x = \pi_1 = \pi_3\), \(y = \pi_2\). From first equation: \(x = \frac{1}{2}y\) From second: \(y = x + x = 2x\) From normalization: \(x + y + x = 1 \implies 2x + y = 1\) Substitute \(y = 2x\): \(2x + 2x = 1 \implies 4x = 1 \implies x = \frac{1}{4}\) Thus, \(y = 2x = \frac{1}{2}\) So, \[ \pi = \left( \frac{1}{4},\ \frac{1}{2},\ \frac{1}{4} \right) \] --- ## 3. Is the chain ergodic? A chain is ergodic if it is irreducible and aperiodic. Check period: - State 1 can return to itself only in an even number of steps (2, 4, ...), because \(1 \to 2 \to 1\) is 2 steps, but not in 1 step. - Same for states 2 and 3. Thus, the chain has period 2 (not aperiodic). **Conclusion:** The chain is **not ergodic**.

# Final Answers 1. **Is the chain irreducible?** - Yes, the chain is irreducible. 2. **Find the stationary distribution, if it exists.** - The stationary distribution is \(\pi = \left( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \right)\). 3. **Is the chain ergodic?** - No, the chain is not ergodic.