Let's go through each part step by step. --- ## 1. **Is the chain irreducible?** **Definition:** A Markov chain is **irreducible** if you can get from any state to any other state (possibly in multiple steps). **Transition matrix:** \[ P = \begin{pmatrix} & 1 & \\ \frac{1}{2} & & \frac{1}{2} \\ & 1 & \end{pmatrix} \] **Check connectivity:** - From **state 1**: Can go to state 2 in 1 step. - From **state 2**: Can go to states 1 or 3 in 1 step. - From **state 3**: Can go to state 2 in 1 step. **Can you reach every state from every other state?** Let's check: - From 1: 1 → 2 (1 step), 1 → 2 → 3 (2 steps) - From 2: 2 → 1 (1 step), 2 → 3 (1 step) - From 3: 3 → 2 (1 step), 3 → 2 → 1 (2 steps) **Conclusion:** Yes, every state communicates with every other state. **The chain is irreducible.** --- ## 2. **Find the stationary distribution** Let \(\pi = (\pi_1, \pi_2, \pi_3)\) be the stationary distribution: \[ \pi P = \pi \] \[ \begin{align*} \pi_1 &= \cdot\pi_1 + \frac{1}{2}\pi_2 + \cdot\pi_3 = \frac{1}{2}\pi_2 \\ \pi_2 &= 1\cdot\pi_1 + \cdot\pi_2 + 1\cdot\pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= \cdot\pi_1 + \frac{1}{2}\pi_2 + \cdot\pi_3 = \frac{1}{2}\pi_2 \\ \pi_1 + \pi_2 + \pi_3 = 1 \end{align*} \] Notice that \(\pi_1 = \pi_3\). Substitute into the normalization: \[ \pi_1 + \pi_2 + \pi_3 = 1 \\ 2\pi_1 + \pi_2 = 1 \] Since \(\pi_1 = \pi_3 = \frac{1}{2}\pi_2\): Let \(\pi_1 = x\), \(\pi_2 = y\), \(\pi_3 = x\): \[ x + y + x = 1 \implies 2x + y = 1 \\ x = \frac{1}{2}y \] So: \[ 2\left(\frac{1}{2}y\right) + y = 1 \implies y + y = 1 \implies y = \frac{1}{2} \] \[ x = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \] **Stationary distribution:** \[ \boxed{ \pi = \left( \frac{1}{4},\ \frac{1}{2},\ \frac{1}{4} \right) } \] --- ## 3. **Is the chain ergodic?** **Definition:** A Markov chain is **ergodic** if it is irreducible and aperiodic. - **Irreducible:** Already shown above. - **Aperiodic:** Check if the period of each state is 1 (i.e., can return to itself in steps with greatest common divisor 1). **Check period:** - For state 1: Only way back to 1 is 1 → 2 → 1 (2 steps), 1 → 2 → 3 → 2 → 1 (4 steps), etc. Only even steps. - For state 2: 2 → 1 → 2 (2 steps), 2 → 3 → 2 (2 steps), etc. - For state 3: Similar logic. So, all returns are in multiples of 2 steps. **GCD is 2, not 1. Thus, the chain is periodic (period 2), NOT aperiodic.** ### **Conclusion:** - The chain is **not ergodic**. --- ## **Summary Table** | Question | Answer | |-------------------------|----------------------------------------| | Irreducible? | Yes | | Stationary Distribution | \(\left(\frac{1}{4},\frac{1}{2},\frac{1}{4}\right)\) | | Ergodic? | No (periodic, period = 2) |