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Unformatted text preview: STAT 724/ECO 761 Spring 2007 Problem Set Four Solutions 1. Consider the symmetric random walk on the integers: S = 0 and if S n = i then the probability is p = 1 / 2 that S n +1 = i + 1 and the probability is q = 1 / 2 that S n +1 = i- 1. We showed in class that this symmetric random walk will eventually re- turn to 0 with probability 1. Compute the expected time it takes to return to 0, and show that it is infinity. I.e. the symmetric random walk, starting at 0, will return to 0 with probability 1, but it will take infinitely long to do so, on average! ( Hint: Use the theorem we proved in class that says F ( s ) = 1- (1- 4 pqs 2 ) 1 / 2 where F ( s ) = ∑ ∞ n =1 f ( n ) s n is the generating function of the sequence de- fined by f ( n ) = P ( S 1 6 = 0 , . . . , S n- 1 6 = 0 , S n = 0).) Solution: The quantity f ( n ) is the probability that the walk returns to 0 on the nth step, for the first time . Thus the expected lenth of time to return to 0 is given by ∑ ∞ n =1 nf ( n ). But this is exactly F (1) since we can differentiate a power series term by term with its radius of convergence. Since we have a formula for F ( s ) we can easily compute its derivative. The result is F ( s ) =- 1 2 (1- 4 pqs 2 )- 1 / 2 (- 8 pqs ) = s √ 1- s 2 since p = 1 / 2 and q = 1 / 2....
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This note was uploaded on 09/24/2009 for the course IE 221 taught by Professor Georgewilson during the Spring '08 term at Lehigh University .
- Spring '08