Suppose that an object can be at any one ofn+1 equally spaced points xq, X

Chapter 7, Problem 13

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Suppose that an object can be at any one ofn+1 equally spaced points xq, X|,... , x. When an object is at location x,-, it is equally likely to move to either x,_i or x,+i and cannot directly move to any other location. Consider the probabilities {P, }"=o that an object starting at location x,- will reach the left endpoint xq before reaching the right endpoint x. Clearly, Pq I and Pn 0. Since the object can move to x, only from x,_| or xI+i and does so with probability \ for each of these locations, P, = if,-. 1 2 for each / = 1, 2,... , n 1. a. Show that 1 0-. 0 1 -h2-. 1_ 2 '. I 0 -k 1 - ' Pi ' r 1 2 P2 0 . Pn-I . . 0 _ b. Solve this system using n 10, 50, and 100. c. Change the probabilities to a and 1 a for movement to the left and right, respectively, and derive the linear system similar to the one in part (a). d. Repeat part (b) with a = |