At all times, an urn contains N balls-some white balls and some black balls. At each stage,
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At all times, an urn contains N balls-some white balls and some black balls. At each stage, a coin having probability p, 0
(a) Is X, n 0) a Markov chain? If so, explain why.
(b) What are its classes? What are their periods? Are they transient or recurrent?
(c) Compute the transition probabilities P.
(d) Let N=2. Find the proportion of time in each state.
(e) Based on your answer in part
(d) and your intuition, guess the answer for the limiting probability in the general case.
(f) Prove your guess in part
(e) either by showing that Equation (4.7) is satisfied or by using the results of Example 4.28. (g) If p 1, what is the expected time until there are only white balls in the urn if initially there are i white and N - i black?
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