# ps5 - UCSB Fall 2009 ECE 235 Problem Set 5 Assigned Friday...

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UCSB Fall 2009 ECE 235: Problem Set 5 Assigned: Friday November 13 Due: Tuesday November 24 (by noon, in course homework box) Reading: Hajek, Chapter 4 and selected parts of Chapter 6; class notes Topics: Independent increments processes; Markov processes Practice problems (not to be turned in): The even numbered problems in Chapter 4, up to Problem 4.34. Ignore all problems (or parts of problems) related to martingales, since we have not covered these. Compare your solutions with the solutions provided to make sure you understand the concepts. Problem 1: Consider the random walk X n = W 1 + ... + W n + X 0 , with { W k } i.i.d. with P [ W k = +1] = p = 1 - P [ W k = - 1]. Set X 0 = 0. Suppose that p < 1 / 2, so that the random walk is drifting downwards. We want to ±nd the probability that the random walk hits a positive level (which should become more and more unlikely, the higher the level). (a) For any positive integer k , let p k = P [ max n 1 X n k ]. Show that p k = p k 1 . Hint:

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ps5 - UCSB Fall 2009 ECE 235 Problem Set 5 Assigned Friday...

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