Pstat160A Winter 2009 Hw #5 Due: Monday , March 2 Problem 1: Short random walk questions. 1) By a very simple conditioning argument and an even simpler recursion, give an alternative method to compute E ( X n ). 2) Hand-out reading questions a) In your own words, what is “path counting” about? b) List all the diﬀerent paths properties considered in the hand-out, and state how many paths have each of those properties. c) List all the probabilities that are computed using path counting arguments d) In the formula for P ( X n = b | X0 = a ), where do the conditions “ n + b-a even” and | b-a | ≤ n come from? e) In the same formula as above, what are the powers of p and q represent respectively? f) What is the reﬂection principle, and what is the main idea that is useful to count paths? g) Give two applications of the reﬂection principle. 3) Link with the Binomial and application to the stock price model a) By going back to the representation in terms of # up and # down, give a representation of
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