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Show that chain in Exercise 1.39 with transition probability is is a special case of the age…


Show that chain in Exercise 1.39 with transition probability is

Show that chain in Exercise 1.39 with transition probability is is a special case of the age...

is a special case of the age chain. Use this observation and the previous exercise to compute the stationary distribution.

Exercise 1.39

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

Show that chain in Exercise 1.39 with transition probability is is a special case of the age...

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?