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145HW3

# 145HW3 - Homework assignment Number 3 Math 145 due on 1...

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Homework assignment Number 3 Math 145 // due on 01/26/2007 1. Problems Section 3.8: 6, 8, 9, 10, 13, 15. 2. A “walk” starts at the origin and makes up (resp. down) steps (1 , 1) (resp. (1 , - 1)). (a) How many walks are there from the origin to ( p, q ) for given integers p > 0, q ? (b) Suppose now that the walk starts at A = (0 , 0) and finishes at B = (2 n, 0). We want to determine the number of such walks that go below the x -axis before 2 n . The aim is to show that this number is equal to the number of walks starting at A and ending at C = (2 n, - 2). i. Let W be a walk going below the x -axis. Suppose that W reaches the level - 1 for the first time at the instant n < 2 n . Set W 1 to be the trajectory which is the reflection with respect to the axis y = - 1 of W . Make a plot of two such trajectories W and W 1 . What are the endpoints of W 1 ? ii. Consider then the walk W which first follows the trajectory of W until time n and then follows the same trajectory as W 1 until time 2 n . Show that W W defines a bijection between the set of walks from A to B going below the
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