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Let X n be the number of days since David last shaved, calculated at 7:30 a.m. when he is trying…

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Let Xn be the number of days since David last shaved, calculated at 7:30 a.m. when he is trying to decide if he wants to shave today. Suppose that Xn is a Markov chain with transition matrix

Let X n be the number of days since David last shaved, calculated at 7:30 a.m. when he is trying...

In words, if he last shaved k days ago, he will not shave with probability 1/(k+1). However, when he has not shaved for 4 days his mother orders him to shave, and he does so with probability 1. (a) What is the long-run fraction of time David shaves? (b) Does the stationary distribution for this chain satisfy the detailed balance condition?