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Imagine a container with a divider in the middle, and a total of balls distributed in some way between the left and right sides. At

Imagine a container with a divider in the middle, and a total of balls distributed in some way between the left and right sides. At each step, one of the m balls is chosen uniformly at random and moved to the other side of the divider (i.e. moved from left to right side, or vice versa). Denote by Xn the number of balls on the left side at time n.

  1. Show that Xn is a Markov Chain, and define its state space.
  2. Calculate the transition probabilities for Xn.
  3. Show that the stationary distribution for this chain is , with the i th entry equal to (i) =2m1(im)

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