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# Assign3 - Introduction to Stochastic Processes Spring 2011...

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Introduction to Stochastic Processes, Spring 2011 Homework #3 Due: Thursday, February 3rd, 2011 in class Simple Random Walk Problems 1. Problem of the Points. A coin is tossed repeatedly, heads turning up with proba- bility p on each toss. Player A wins the game if m heads appear before n tails have appeared, and player B wins if n tails appear before m heads. Let p mn be the proba- bility that A wins the game. Set up a difference equation for the p mn . What are the boundary conditions? 2. Let S n be a simple random walk with S 0 = 0, i.e. S n = n X i =1 X i where X 1 , X 2 , X 3 , . . . are independent random variables with P ( X i = 1) = p and P ( X i = - 1) = 1 - p = q . (a) Note that at odd times S n is necessarily odd, and at even times it is even. Hence it can only be back at zero at even times. For n 0, compute P ( S 2 n = 0). Hint: In order to be back at zero at time 2 n , you need to have made as many “up” steps as “down” steps. What is the probability of that event?

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