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4.2.5 Suppose that the weather on any day depends on the weather conditions during the previous 2 days. We form a Markov chain with the following states: State (S, S) if it was sunny both today and yesterday, State (S,C) if it was sunny yesterday but cloudy today, State (C,S) if it was cloudy yesterday but sunny today, State (C, C) if it was cloudy both today and yesterday, and transition probability matrix Today's state (S,S) (S, C) (C,S) (C,C) (S, S) 0.7 0.3 0 0 (S, C) 0 0 0.4 0.6 P = (C,S) 0.5 0.5 0 0 (C, C) 0 0 0.2 0.8 (a) Given that it is sunny on days 0 and 1, what is the probability it is sunny on day 5? (b) In the long run, what fraction of days are sunny?

4.2.5 Suppose that the weather on any day depends on the weather conditions during the previous 2 days. We form a Markov chain with the following states: State (S, S) if it was sunny both today and yesterday, State (S,C) if it was sunny yesterday but cloudy today, State (C,S) if it was cloudy yesterday but sunny today, State (C, C) if it was cloudy both today and yesterday, and transition probability matrix Today's state (S,S) (S, C) (C,S) (C,C) (S, S) 0.7 0.3 0 0 (S, C) 0 0 0.4 0.6 P = (C,S) 0.5 0.5 0 0 (C, C) 0 0 0.2 0.8 (a) Given that it is sunny on days 0 and 1, what is the probability it is sunny on day 5? (b) In the long run, what fraction of days are sunny?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 9EQ
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4.2.5 Suppose that the weather on any day depends on the weather conditions during the previous 2 days. We form a Markov chain with the following states: State (S, S) if it was sunny both today and yesterday, State (S,C) if it was sunny yesterday but cloudy today, State (C,S) if it was cloudy yesterday but sunny today, State (C, C) if it was cloudy both today and yesterday, and transition probability matrix Today's state (S,S) (S, C) (C,S) (C,C) (S, S) 0.7 0.3 0 0 (S, C) 0 0 0.4 0.6 P = (C,S) 0.5 0.5 0 0 (C, C) 0 0 0.2 0.8 (a) Given that it is sunny on days 0 and 1, what is the probability it is sunny on day 5? (b) In the long run, what fraction of days are sunny?
Transcribed Image Text:4.2.5 Suppose that the weather on any day depends on the weather conditions during the previous 2 days. We form a Markov chain with the following states: State (S, S) if it was sunny both today and yesterday, State (S,C) if it was sunny yesterday but cloudy today, State (C,S) if it was cloudy yesterday but sunny today, State (C, C) if it was cloudy both today and yesterday, and transition probability matrix Today's state (S,S) (S, C) (C,S) (C,C) (S, S) 0.7 0.3 0 0 (S, C) 0 0 0.4 0.6 P = (C,S) 0.5 0.5 0 0 (C, C) 0 0 0.2 0.8 (a) Given that it is sunny on days 0 and 1, what is the probability it is sunny on day 5? (b) In the long run, what fraction of days are sunny?
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