Alternative derivation for the probabilities for the number of changes of sign. Show that Er. 2n-1 = 1 2 k=1 [5-1.2m-1-2x + 5.2n-1-2k]. Assuming by induction that (5.1) holds for all epochs prior to 2n - 1 show that this reduces to = Gr.2n-1-fax Pan.2r which is the probability of reaching the point (2n, 2r) after a visit to the origin. Using the ballot theorem conclude that (5.1) holds.