Example 4.3 On any given day Gary is either cheerful (C), so-so (S), or glum

(G). If he is cheerful today, then he will be C, S, or G tomorrow with respective probabilities 0.5, 0.4, 0.1. If he is feeling so-so today, then he will be C, S, or G tomorrow with probabilities 0.3, 0.4, 0.3. If he is glum today, then he will be C, S, or G tomorrow with probabilities 0.2, 0.3, 0.5.

Letting Xn denote Gary’s mood on the nth day, then {Xn, n 0} is a three-state Markov chain (state 0 = C, state 1 = S, state 2 = G) with transition probability matrix

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0.5 P=0.3 0.4 0.11 0.4 0.3 0.2 0.3 0.5