# Question

Suppose a Bernoulli trial results in a success with probability p and a failure with probability 1 – p. Suppose the Bernoulli trial is repeated indefinitely with each repitition independent of all others. Let Xn be a “ success runs” Markov chain where represents the number of most recent consecutive successes that have been observed at the n th trial. That is Xn = m, if trial numbers n, n– 1, n – 2,…, n– m+ 1 were all successes but trial number n–m was a failure. Note that Xn = 0 if the th trial was a failure.

(a) Find an expression for the one- step transition probabilities, p i, j.

(b) Find an expression for the - step first return probabilities for state 0,f0(n)0 .

(c) Prove that state 0 is recurrent for any 0 < p < 1. Note that since all states communicate with one another, this result together with the result of Exercise 9.29 is sufficient to show that all states are recurrent.

(a) Find an expression for the one- step transition probabilities, p i, j.

(b) Find an expression for the - step first return probabilities for state 0,f0(n)0 .

(c) Prove that state 0 is recurrent for any 0 < p < 1. Note that since all states communicate with one another, this result together with the result of Exercise 9.29 is sufficient to show that all states are recurrent.

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