Let's go through each part step by step: --- ## 1. **Is the chain irreducible?** A Markov chain is **irreducible** if it is possible to get from any state to any other state (possibly in multiple steps). Given the transition matrix: \[ P = \begin{pmatrix} & 1 & \\ \frac{1}{2} & & \frac{1}{2} \\ & 1 & \end{pmatrix} \] Let's label the states as 1, 2, 3: - **From 1:** Can go to 2 (in 1 step), from 2 to 1 (in 1 step), so can return. - **From 1 to 3:** From 1 → 2 (step 1), then 2 → 3 (step 2), so possible in 2 steps. - **From 3 to 1:** 3 → 2 (step 1), then 2 → 1 (step 2), possible in 2 steps. - **Any pair can reach each other in at most 2 steps.** **Conclusion:** **Yes, the chain is irreducible.** --- ## 2. **Find the stationary distribution, if it exists.** The **stationary distribution** \(\pi = (\pi_1, \pi_2, \pi_3)\) satisfies: \[ \pi P = \pi \] and \(\pi_1 + \pi_2 + \pi_3 = 1\). Set up the equations: \[ \begin{align*} \pi_1 &= \cdot \pi_1 + \frac{1}{2} \pi_2 + \cdot \pi_3 = \frac{1}{2}\pi_2 \\ \pi_2 &= 1 \cdot \pi_1 + \cdot \pi_2 + 1 \cdot \pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= \cdot \pi_1 + \frac{1}{2} \pi_2 + \cdot \pi_3 = \frac{1}{2}\pi_2 \\ \end{align*} \] Notice that \(\pi_1 = \pi_3\). Also, \(\pi_2 = \pi_1 + \pi_3 = 2\pi_1\). Now, normalization: \[ \pi_1 + \pi_2 + \pi_3 = 1 \\ \pi_1 + 2\pi_1 + \pi_1 = 1 \\ 4\pi_1 = 1 \implies \pi_1 = \frac{1}{4} \] So, \[ \pi_1 = \pi_3 = \frac{1}{4},\ \pi_2 = \frac{1}{2} \] **Stationary distribution:** \[ \boxed{\left(\frac{1}{4},\, \frac{1}{2},\, \frac{1}{4} \right)} \] --- ## 3. **Is the chain ergodic?** A Markov chain is **ergodic** if it is both irreducible and aperiodic. - We already showed it is **irreducible**. - **Aperiodicity:** Look at the period of state 2, for example. You can only return to state 2 in an even number of steps (since it goes 2 → 1 or 3 → 2). The period is 2. The same for states 1 and 3. So, all states have period 2. **Conclusion:** **The chain is *not* ergodic, because it is not aperiodic (it is periodic with period 2).** --- ## **Summary Table** | Property | Answer | |---------------------|----------------------------------------------------------------------------| | Irreducible? | Yes | | Stationary Dist.? | \(\left(\frac{1}{4},\, \frac{1}{2},\, \frac{1}{4} \right)\) | | Ergodic? | No, because it is periodic (period 2), not aperiodic |