Let's break down the solution for the Markov chain step by step, using clear definitions and detailed calculations. --- # 1. Is the Chain Irreducible? ### **Definition** A Markov chain is **irreducible** if it is possible to reach any state from any other state (possibly in multiple steps). ### **Transition Matrix** \[ P = \begin{pmatrix} & 1 & \\ \frac{1}{2} & & \frac{1}{2} \\ & 1 & \end{pmatrix} \] ### **Connectivity Check** - **From State 1:** Can move to State 2 in one step. - **From State 2:** Can move to State 1 or State 3 in one step. - **From State 3:** Can move to State 2 in one step. - **In two steps:** - From State 1: 1 → 2 → 1 or 1 → 2 → 3 (so from 1 you can get to 3 in two steps) - From State 3: 3 → 2 → 1 (from 3 you can get to 1 in two steps) ### **Conclusion** **Yes, the chain is irreducible** because you can reach any state from any other state. --- # 2. Find the Stationary Distribution ### **Definition** A stationary distribution \( \pi = (\pi_1, \pi_2, \pi_3) \) satisfies: \[ \pi P = \pi, \quad \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Set Up the Equations** From \( \pi P = \pi \): \[ \begin{align*} \pi_1 &= \cdot \pi_1 + \frac{1}{2} \cdot \pi_2 + \cdot \pi_3 = \frac{1}{2} \pi_2 \\ \pi_2 &= 1 \cdot \pi_1 + \cdot \pi_2 + 1 \cdot \pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= \cdot \pi_1 + \frac{1}{2} \cdot \pi_2 + \cdot \pi_3 = \frac{1}{2} \pi_2 \end{align*} \] And the normalization condition: \[ \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Solving the Equations** From above: - \( \pi_1 = \frac{1}{2} \pi_2 \) - \( \pi_3 = \frac{1}{2} \pi_2 \) - \( \pi_2 = \pi_1 + \pi_3 \) Substitute \( \pi_1 \) and \( \pi_3 \) in terms of \( \pi_2 \): \[ \pi_2 = \frac{1}{2} \pi_2 + \frac{1}{2} \pi_2 = \pi_2 \] (This is always true.) Now use the normalization: \[ \pi_1 + \pi_2 + \pi_3 = 1 \\ \frac{1}{2} \pi_2 + \pi_2 + \frac{1}{2} \pi_2 = 1 \\ 2 \pi_2 = 1 \\ \pi_2 = \frac{1}{2} \] Calculate \( \pi_1 \) and \( \pi_3 \): \[ \pi_1 = \frac{1}{2} \pi_2 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \\ \pi_3 = \frac{1}{2} \pi_2 = \frac{1}{4} \] ### **Final 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. #### **Aperiodicity Check** A state \( i \) is aperiodic if you can return to it at irregular intervals. A chain is aperiodic if all states are aperiodic. - **State 1:** Can only return in even steps (2, 4, ...), because 1→2→1 (2 steps). - **State 3:** Same as State 1. - **State 2:** Can return to itself in 2 steps (2→1→2, 2→3→2), but never in 1 step. **All states have period 2 (since you can only return in even steps), so the chain is periodic, not aperiodic.** ### **Conclusion** **The chain is NOT ergodic** because it is not aperiodic, even though it is irreducible. --- ## **Summary Table** | Property | Result | |---------------------|--------------------------| | Irreducible | Yes | | Stationary Dist. | \( \left(\frac{1}{4}, \frac{1}{2}, \frac{1}{4}\right) \) | | Ergodic | No (period = 2) |

# Analysis of the Markov Chain Let's analyze the given Markov chain step by step, using clear definitions and detailed calculations. --- ## Transition Matrix ### **Given Transition Matrix** The transition matrix is: \[ P = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \] --- ## 1. Is the Chain Irreducible? ### **Definition** 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 ### **Definition** A stationary distribution \( \pi = (\pi_1, \pi_2, \pi_3) \) satisfies: \[ \pi P = \pi, \quad \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Set Up the Equations** From \( \pi P = \pi \): \[ \begin{align*} \pi_1 &= 0 \cdot \pi_1 + 1 \cdot \pi_2 + 0 \cdot \pi_3 = \pi_2 \\ \pi_2 &= 1 \cdot \pi_1 + 0 \cdot \pi_2 + 1 \cdot \pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= 0 \cdot \pi_1 + 1 \cdot \pi_2 + 0 \cdot \pi_3 = \pi_2 \end{align*} \] ### **Normalization Condition** \[ \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Solving the Equations** From the equations: 1. \( \pi_1 = \pi_2 \) 2. \( \pi_3 = \pi_2 \) 3. Substitute into the normalization equation: \[ \pi_2 + \pi_2 + \pi_2 = 1 \implies 3 \pi_2 = 1 \implies \pi_2 = \frac{1}{3} \] 4. Therefore: \[ \pi_1 = \frac{1}{3}, \quad \pi_3 = \frac{1}{3} \] ### **Final Stationary Distribution** \[ \boxed{\pi = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)} \] --- ## 3. Is the Chain Ergodic? ### **Definition** A Markov chain is **ergodic** if it is irreducible and aperiodic. ### **Aperiodicity Check** - **State 1:** Can return to itself in 2 steps (1 → 2 → 1). - **State 2:** Can return to itself in 2 steps (2 → 1 → 2, or 2 → 3 → 2). - **State 3:** Can return to itself in 2 steps (3 → 2 → 3). All states return in even steps, which indicates that the chain is periodic with period 2. ### **Conclusion** **The chain is NOT ergodic** because it is not aperiodic. --- ## **Summary Table** | Property | Result | |---------------------|--------------------------| | Irreducible | Yes | | Stationary Dist. | \( \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \) | | Ergodic | No (period = 2) |

# Analysis of the Markov Chain Let's analyze the given Markov chain step by step, using clear definitions and detailed calculations. --- ## Transition Matrix ### **Given Transition Matrix** The transition matrix is: \[ P = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \] --- ## 1. Is the Chain Irreducible? ### **Definition** 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 ### **Definition** A stationary distribution \( \pi = (\pi_1, \pi_2, \pi_3) \) satisfies: \[ \pi P = \pi, \quad \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Set Up the Equations** From \( \pi P = \pi \): \[ \begin{align*} \pi_1 &= 0 \cdot \pi_1 + 1 \cdot \pi_2 + 0 \cdot \pi_3 = \pi_2 \\ \pi_2 &= 1 \cdot \pi_1 + 0 \cdot \pi_2 + 1 \cdot \pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= 0 \cdot \pi_1 + 1 \cdot \pi_2 + 0 \cdot \pi_3 = \pi_2 \end{align*} \] ### **Normalization Condition** \[ \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Solving the Equations** 1. From the first equation: - \( \pi_1 = \pi_2 \) 2. From the third equation: - \( \pi_3 = \pi_2 \) 3. Substitute into the normalization equation: \[ \pi_2 + \pi_2 + \pi_2 = 1 \implies 3 \pi_2 = 1 \implies \pi_2 = \frac{1}{3} \] 4. Therefore: \[ \pi_1 = \frac{1}{3}, \quad \pi_3 = \frac{1}{3} \] ### **Final Stationary Distribution** \[ \boxed{\pi = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)} \] --- ## 3. Is the Chain Ergodic? ### **Definition** A Markov chain is **ergodic** if it is irreducible and aperiodic. ### **Aperiodicity Check** - **State 1:** Can return to itself in 2 steps (1 → 2 → 1). - **State 2:** Can return to itself in 2 steps (2 → 1 → 2, or 2 → 3 → 2). - **State 3:** Can return to itself in 2 steps (3 → 2 → 3). All states return in even steps, indicating that the chain is periodic with period 2. ### **Conclusion** **The chain is NOT ergodic** because it is not aperiodic. --- ## **Summary Table** | Property | Result | |---------------------|--------------------------| | Irreducible | Yes | | Stationary Dist. | \( \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \) | | Ergodic | No (period = 2) |

# Analysis of the Markov Chain Let's analyze the given Markov chain step by step, using clear definitions and detailed calculations. --- ## Transition Matrix ### **Given Transition Matrix** The transition matrix is: \[ P = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \] --- ## 1. Is the Chain Irreducible? ### **Definition** 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 ### **Definition** A stationary distribution \( \pi = (\pi_1, \pi_2, \pi_3) \) satisfies: \[ \pi P = \pi, \quad \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Set Up the Equations** From \( \pi P = \pi \): \[ \begin{align*} \pi_1 &= 0 \cdot \pi_1 + 1 \cdot \pi_2 + 0 \cdot \pi_3 = \pi_2 \\ \pi_2 &= 1 \cdot \pi_1 + 0 \cdot \pi_2 + 1 \cdot \pi_3 = \pi_1 + \pi_3 \\ \pi_3 &= 0 \cdot \pi_1 + 1 \cdot \pi_2 + 0 \cdot \pi_3 = \pi_2 \end{align*} \] ### **Normalization Condition** \[ \pi_1 + \pi_2 + \pi_3 = 1 \] ### **Solving the Equations** From the equations: 1. \( \pi_1 = \pi_2 \) 2. \( \pi_3 = \pi_2 \) 3. Substitute into the normalization equation: \[ \pi_2 + \pi_2 + \pi_2 = 1 \implies 3 \pi_2 = 1 \implies \pi_2 = \frac{1}{3} \] 4. Therefore: \[ \pi_1 = \frac{1}{3}, \quad \pi_3 = \frac{1}{3} \] ### **Final Stationary Distribution** \[ \boxed{\pi = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)} \] --- ## 3. Is the Chain Ergodic? ### **Definition** A Markov chain is **ergodic** if it is irreducible and aperiodic. ### **Aperiodicity Check** - **State 1:** Can return to itself in 2 steps (1 → 2 → 1). - **State 2:** Can return to itself in 2 steps (2 → 1 → 2, or 2 → 3 → 2). - **State 3:** Can return to itself in 2 steps (3 → 2 → 3). All states return in even steps, indicating that the chain is periodic with period 2. ### **Conclusion** **The chain is NOT ergodic** because it is not aperiodic. --- ## **Summary Table** | Property | Result | |---------------------|--------------------------| | Irreducible | Yes | | Stationary Dist. | \( \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \) | | Ergodic | No (period = 2) |