Four balls are distributed between bucket 1 and bucket 2. Each step. a random ball is...

  

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Four balls are distributed between bucket 1 and bucket 2. Each step. a random ball is chosen, and moved to a different bucket. (a) Draw the directed graph (with arc-probabilities) describing the relevant discrete Markov chain. (b) Write down the transition matrix P of the Markov chain. (c) Find the probability that after exactly 3 steps, starting with 2 balls in each bucket, we now have all balls in bucket 1. (d) List the communication classes of the Markov chain. For each class, determine whether the class is recurrent or transient, and whether the class is closed. (e) Find a stationary distribution of the Markov chain. Can we be sure that it is unique? Four balls are distributed between bucket 1 and bucket 2. Each step. a random ball is chosen, and moved to a different bucket. (a) Draw the directed graph (with arc-probabilities) describing the relevant discrete Markov chain. (b) Write down the transition matrix P of the Markov chain. (c) Find the probability that after exactly 3 steps, starting with 2 balls in each bucket, we now have all balls in bucket 1. (d) List the communication classes of the Markov chain. For each class, determine whether the class is recurrent or transient, and whether the class is closed. (e) Find a stationary distribution of the Markov chain. Can we be sure that it is unique?

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