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At all times, an urn contains N balls&#8211;some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing...

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At all times, an urn contains N balls&#8211;some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing heads is flipped. If heads appears, then a ball is chosen at random from the urn and is replaced by a white ball. Let Xn denote the number of white balls in the urn after the nth stage.

(a)Is {Xn , n &#8805; 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 Pi,j .
(d) Let N = 2. Find the proportion of time in each state.
(f) Prove your guess in part (c)
(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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