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Determine the matrix of the transition probabilities

Question:

Weather is classified as (predominantly) sunny (S) and (predominantly) cloudy (C), where C includes rain. For the town of Musi, a fairly reliable prediction of tomorrow's weather can only be made on the basis of today's and yesterday's weather. Let (C,S) indicate that the weather yesterday was cloudy and today's weather is sunny and so on. Based on historical observations it is known that, given the constellation (S,S) today, the weather tomorrow will be sunny with probability 0.8 and cloudy with probability 0.2; given (S,C) today, the weather tomorrow will be sunny with probability 0.4 and cloudy with probability 0.6; given (C,S) today, the weather tomorrow will be sunny with probability 0.6 and cloudy with probability 0.4; given (C,C) today, the weather tomorrow will be cloudy with probability 0.8 and sunny with probability 0.2.

(1) Illustrate graphically the transitions between the states

1= (S,S), 2 = (S,C), 3 = (C,S), and 4 = (C,C)

(2) Determine the matrix of the transition probabilities of the corresponding discretetime Markov chain and its stationary state distribution.

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