Suppose that the weather in a particular region behaves according to a Markov chain. Specifically, suppose that the probability that tomorrow will be a wet day is 0.662 if today is wet and 0.250 if today is dry. The probability that tomorrow will be a dry day is 0.750 if today is dry and 0.338 if today is wet. [This exercise is based on an actual study of rainfall in Tel Aviv over a 27-year period. See K. R. Gabriel and J. Neumann, “A Markov Chain Model for Daily Rainfall Occurrence at Tel Aviv,” Quarterly Journal of the Royal Meteorological Society, 88 (1962), pp. 90–95.]
(a) Write down the transition matrix for this Markov chain.
(b) If Monday is a dry day, what is the probability that Wednesday will be wet?
(c) In the long run, what will the distribution of wet and dry days be?
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