Question 3: (Transforming a process into a Markov Chain) Suppose that whether or not it rains today depends on previous

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Question 3 Transforming A Process Into A Markov Chain Suppose That Whether Or Not It Rains Today Depends On Previous 1 (73.94 KiB) Viewed 30 times

Question 3: (Transforming a process into a Markov Chain) Suppose that whether or not it rains today depends on previous weather conditions through the last two days. Specifically, suppose that if it has rained for the past two days, then it will rain tomorrow with probability 0.6; if it rained today but not yesterday, then it will rain tomorrow with probability 0.5; if it rained yesterday but not today, then it will rain tomorrow with probability 0.4; if it has not rained in the past two days, then it will rain tomorrow with probability 0.2. If we let the state at time n depend only on whether or not it is raining at time n, then the preceding model is not a Markov chain. However, we can transform this model into a Markov chain by saying that the state at any time is determined by the weather conditions during both that day and the previous day. In other words, we can say that the process is in state 0 if it rained both today and yesterday, state 1 if it rained today but not yesterday, state 2 if it rained yesterday but not today, state 3 if it did not rain either yesterday or today. The preceding would then represent a four-state Markov chain having a transition probability matrix 0.6 0 0.4 0 0.5 0 0.5 0 P = 0 0.4 0 0.6 0 0.2 0 0.8 1. Draw the transition diagram of this chain. 2. This chain is it irreducible? 3. Determine which states are transient and which are recurrent. 4. Find the two-step transition matrix 5. Given that it rained on Monday, what is the probability that it will rain on Thursday? 6. Given that it rained on Monday and Tuesday, what is the probability that it will rain on Thursday?