On a given day, Jeff is either cheerful, so-so, or glum. Given that he is cheerful on a given day, then he will be cheerful the next day with probability 0.6, so-so with probability 0.2, and glum with probability 0.2. Given that he is so-so on a given day, then he will be cheerful the next day with probability 0.3, so-so with probability 0.5, and glum with probability 0.2. Given that he is glum on a given day, then he will be so-so the next day with probability 0.5,and glum with the probability 0.5. Let Xn denote Jeff's mood on the nth day. Show that {Xn: n=0,1,2} is a Markov chain.
(a) Give the state transition probability matrix
(b) Given that Jeff was so-so on Monday, what is the probability that he will be cheerful on Wednesday and Friday and glum on Sunday?
(c) On the long run, what proportion of time is Jeff in each of the 3 moods?