1. Quiz question Two dogs, Hinkler and Moke, have three fleas between them. Every day, one (and only one) flea jumps from the dog it is currently on and lands on the other dog. On each day, which flea jumps is random, and all three fleas have the same probability of jumping, regardless of which dogs the fleas are on. Define Xn to be the number of fleas on Moke after n days. Then (X,, n = 0, 1, 2, ... ) forms a Markov chain with state space S = {0, 1, 2, 3}. Initially, all three fleas are on Hinkler, meaning that Xo = 0. (a) Some of the entries of the transition matrix, P, are shown. Find the rest. 0 0 0 P = 0 2/3 0 1/3 (b) Confirm to yourself that this Markov chain is a birth-death chain. (c) Find the equilibrium distribution(s) * = (70, 71, #2, #3). Is there a unique equilibrium distribution, or more than one? Hint: use the Detailed Balance Equations