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Suppose a random walk on S = {0, 1, 2, 3, 4} works as follows. If...

Suppose a random walk on S = {0, 1, 2, 3, 4} works as follows. If 1 ≤ k ≤ 3 and Xn = k, then Xn+1 is k ± 1 each with probability 1/2. If Xn = 0, then Xn+1 = 0 (0 is absorbing). If Xn = 4, then Xn+1 is either 4 or 3, each with probability 1/2 (4 is “retaining”). Let T = min{n ≥ 0 : Xn = 0} denote the time that the walk gets absorbed at 0. For each k with 0 ≤ k ≤ 4, determine τk = E[T|X0 = k]. (E.g., τ0 = 0.)

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