chap2 34 ross - STAT 724/ECO 761 Spring 2007 Problem Set...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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 .

Page1 / 5

chap2 34 ross - STAT 724/ECO 761 Spring 2007 Problem Set...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online