Let's tackle each question step by step. --- ## 1. **Is the chain irreducible?** A **Markov chain** is **irreducible** if you can 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, and 3. - **From state 1:** - Can go to 2 in one step (since \( P_{1,2} = 1 \)) - **From state 2:** - Can go to 1 or 3 in one step (\( P_{2,1} = \frac{1}{2} \), \( P_{2,3} = \frac{1}{2} \)) - **From state 3:** - Can go to 2 in one step (\( P_{3,2} = 1 \)) Let's check connectivity: - From 1 → 2 → 3 (in 2 steps: 1→2, 2→3) - From 3 → 2 → 1 (in 2 steps: 3→2, 2→1) - And so on. **Conclusion:** **Yes, the chain is irreducible** because you can get from any state to any other state. --- ## 2. **Find the stationary distribution \(\pi\), if it exists.** The **stationary distribution** \(\pi = (\pi_1, \pi_2, \pi_3)\) satisfies: \[ \pi P = \pi, \quad \text{and} \quad \pi_1 + \pi_2 + \pi_3 = 1 \] Write the equations: \[ \begin{align*} \pi_1 &= \cdot \pi_1 + \frac{1}{2} \pi_2 + \cdot \pi_3 \\ \pi_2 &= 1 \cdot \pi_1 + \cdot \pi_2 + 1 \cdot \pi_3 \\ \pi_3 &= \cdot \pi_1 + \frac{1}{2} \pi_2 + \cdot \pi_3 \end{align*} \] Which simplifies to: \[ \begin{align*} \pi_1 &= \frac{1}{2} \pi_2 \\ \pi_2 &= \pi_1 + \pi_3 \\ \pi_3 &= \frac{1}{2} \pi_2 \\ \pi_1 + \pi_2 + \pi_3 &= 1 \end{align*} \] But \(\pi_1 = \pi_3 = \frac{1}{2}\pi_2\), so: \[ \pi_2 = \pi_1 + \pi_3 = 2 \cdot \pi_1 \] But also, from above, \(\pi_1 + \pi_2 + \pi_3 = 1\): \[ \pi_1 + \pi_2 + \pi_3 = \pi_1 + 2\pi_1 + \pi_1 = 4\pi_1 = 1 \implies \pi_1 = \frac{1}{4} \] \[ \pi_2 = 2\pi_1 = \frac{1}{2} \] \[ \pi_3 = \pi_1 = \frac{1}{4} \] **Stationary distribution:** \[ \boxed{ \pi = \left( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \right) } \] --- ## 3. **Is the chain ergodic?** A Markov chain is **ergodic** if it is: - **Irreducible** (which we've shown) - **Aperiodic** (the system does not get trapped in cycles of fixed length) Let's check **aperiodicity**: - From state 1, you can only go to 2 in one step. - From state 2, you can go to 1 or 3 in one step. - From state 3, you can only go to 2 in one step. However: - From 1, you can only return to 1 in an even number of steps (since you must go 1→2→1, which takes 2 steps, or 1→2→3→2→1, which takes 4 steps, etc.). - Similarly for state 3. Thus, the **period** of state 1 is 2 (since you can return only in even steps). Likewise, for state 3. **Conclusion:** **The chain is not aperiodic, so it is not ergodic.** --- ## **Summary Table** | Property | Result | |--------------------|------------------------------------------------| | Irreducible? | Yes | | Stationary Dist.? | \( \left( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \right) \) | | Ergodic? | **No** (it is periodic, not aperiodic) | --- If you have questions about any of the steps, let me know!

# Markov Chains Analysis Let's break down the analysis of the given Markov chain step by step. ## Transition Matrix The transition matrix is given as: \[ P = \begin{pmatrix} 0 & 1 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \end{pmatrix} \] ## 1. Is the Chain Irreducible? A Markov chain is **irreducible** if it is possible to reach any state from any other state. ### Connectivity Check: - **From State 1:** - Can move to State 2. - **From State 2:** - Can move to State 1 or State 3. - **From State 3:** - Can move to State 2. ### Conclusion: You can reach any state from any other state. Therefore, **the chain is irreducible**. --- ## 2. Find the Stationary Distribution The stationary distribution \(\pi = (\pi_1, \pi_2, \pi_3)\) satisfies: \[ \pi P = \pi \quad \text{and} \quad \pi_1 + \pi_2 + \pi_3 = 1 \] ### Setting Up Equations: From \(\pi P = \pi\): 1. \(\pi_1 = \frac{1}{2} \pi_2\) 2. \(\pi_2 = \pi_1 + \pi_3\) 3. \(\pi_3 = \frac{1}{2} \pi_2\) ### Normalization Condition: \[ \pi_1 + \pi_2 + \pi_3 = 1 \] ### Solving the Equations: Substituting \(\pi_1\) and \(\pi_3\) in terms of \(\pi_2\): - From (1): \(\pi_1 = \frac{1}{2} \pi_2\) - From (3): \(\pi_3 = \frac{1}{2} \pi_2\) Substituting into the normalization condition: \[ \frac{1}{2} \pi_2 + \pi_2 + \frac{1}{2} \pi_2 = 1 \] \[ 2 \pi_2 = 1 \implies \pi_2 = \frac{1}{2} \] Then, substituting back to find \(\pi_1\) and \(\pi_3\): - \(\pi_1 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\) - \(\pi_3 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\) ### Stationary Distribution: \[ \boxed{\pi = \left( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \right)} \] --- ## 3. Is the Chain Ergodic? A Markov chain is **ergodic** if it is irreducible and **aperiodic**. ### Checking Aperiodicity: - From State 1, you can return to State 1 in 2 steps (1→2→1). - From State 3, you can return to State 3 in 2 steps (3→2→3). Since the only returns to State 1 and State 3 occur in even steps, the period is 2. ### Conclusion: The chain is not aperiodic because it cycles between states in fixed intervals of 2. Thus, **the chain is not ergodic**. --- ## Summary - **Irreducible:** Yes - **Stationary Distribution:** \( \left( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \right) \) - **Ergodic:** No (it is periodic) If further clarification on any part is needed, feel free to ask!