Quiz Question Consider the Markov chain with transition diagram345210.10.50.50.60.30.50.20.40.30.50.50.20.4i.e., with transition matrixP =0.6 0.4 0 0 00.5 0.5 0 0 00 0.5 0 0.5 00 0.5 0.3 0 0.20 0.1 0.2 0.4 0.3 .(a) Starting from state 5, what is the most likely two-step path? What is the probability(starting from state 5) of following that two-step path?(b) In this Markov chain, can you get from every state to every other state eventually?
Question
Quiz Question Consider the Markov chain with transition diagram345210.10.50.50.60.30.50.20.40.30.50.50.20.4i.e., with transition matrixP =0.6 0.4 0 0 00.5 0.5 0 0 00 0.5 0 0.5 00 0.5 0.3 0 0.20 0.1 0.2 0.4 0.3 .(a) Starting from state 5, what is the most likely two-step path? What is the probability(starting from state 5) of following that two-step path?(b) In this Markov chain, can you get from every state to every other state eventually?
Solution
(a) The most likely two-step path starting from state 5 can be determined by looking at the transition probabilities from state 5. From state 5, we can either go to state 2, 3, or 4. The transition probabilities are 0.1, 0.2, and 0.4 respectively. So, the most likely first step is to go to state 4. From state 4, we can either go to state 2 or 3. The transition probabilities are 0.5 and 0.3 respectively. So, the most likely second step is to go to state 2. Therefore, the most likely two-step path is 5 -> 4 -> 2. The probability of this path is the product of the transition probabilities, which is 0.4 * 0.5 = 0.2.
(b) In this Markov chain, you cannot get from every state to every other state eventually. For example, from state 1, you can only go to state 2 and then back to state 1. You cannot reach states 3, 4, or 5 from state 1. Similarly, from state 3, you can only go to states 2 and 4, but not to states 1 or 5. Therefore, this Markov chain is not ergodic.
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