Question: Random replacement I. Consider an urn containing initially N balls. Let Xn be the number of balls in the urn after performing the following procedure

Random replacement I. Consider an urn containing initially N balls. Let Xn be the number of balls in the urn after performing the following procedure n times. If the urn is non-empty, one of the balls is removed at random; by flipping a fair coin, it is then decided whether or not the ball is returned to the urn. If the urn is empty, the fair coin is used to decide whether or not the urn is filled afresh with N balls. Describe this situation as a Markov chain and find the transition matrix. What is the distribution of Xn as n!1?

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