=+18. Consider a random walk on the integers 0,...,m with transition probabilities pij =

* qi j = i − 1 1 − qi j = i + 1 for i = 1,...,m − 1 and p00 = pmm = 1. All other transition probabilities are 0. Eventually the walk gets trapped at 0 or m. Let fi be the probability that the walk is absorbed at 0 starting from i. Show that fi is an increasing function of the entries of q = (q1,...,qm−1).

(Hint: Let q and q∗ satisfy qi ≤ q∗

i for i = 1,...,m − 1. Construct