4.10 On a given day Mark is cheerful, so-so, or glum. Given that he is cheerful on a given day, then he will be cheerful again 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 again 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 again with probability 0.5. Let state 1 denote the cheerful state, state 2 denote the so-so state, and state 3 denote the glum state. Let Xn denote Mark’s mood on the nth day, then fXn; n 5 0; 1; 2; ...g is a threestate Markov chain.

a. Draw the state-transition diagram of the process.

b. Give the state-transition probability matrix.

c. Given that Mark was so-so on Monday, what is the probability that he will be cheerful on Wednesday and Friday and glum on Sunday?

d. On the long run, what proportion of time is Mark in each of his three moods?