For simple random walk Sn with absorbing barriers at 0 and N, let W be the...

  

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For simple random walk Sn with absorbing barriers at 0 and N, let W be the event that the particle is absorbed at 0 rather than at N, and let Pk = P (W|So = k). Show that, if the particle starts at k with 0 <k < N, the conditional probability that the first step is rightwards, given W, equals PPk+¹. Deduce that the expected duration of the walk, conditional on W when So = k, satisfies the PR equation PPk+1Jk+1-Pk Jk+ (Pk - PPk+1) Jk-1 = -Pk. 0 <k < N. Take Jo = 0 as a boundary condition. Find Jk in the symmetric case p = 1/2. Hint. Let Y, be the duration of the walk (number of steps until absorption) when So = k. Then JkE (YkW), and Jk = E(Yk|1st step rightwards, given W) P (1st step rightwards W) +E (Yk|1st step leftwards, given W) P (1st step lefttwards|W). For simple random walk Sn with absorbing barriers at 0 and N, let W be the event that the particle is absorbed at 0 rather than at N, and let Pk = P (W|So = k). Show that, if the particle starts at k with 0 <k < N, the conditional probability that the first step is rightwards, given W, equals PPk+¹. Deduce that the expected duration of the walk, conditional on W when So = k, satisfies the PR equation PPk+1Jk+1-Pk Jk+ (Pk - PPk+1) Jk-1 = -Pk. 0 <k < N. Take Jo = 0 as a boundary condition. Find Jk in the symmetric case p = 1/2. Hint. Let Y, be the duration of the walk (number of steps until absorption) when So = k. Then JkE (YkW), and Jk = E(Yk|1st step rightwards, given W) P (1st step rightwards W) +E (Yk|1st step leftwards, given W) P (1st step lefttwards|W).

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To solve this problem lets denote by Pk the conditional probability that the first step is rightwards given that the particle is absorbed at 0 rather than at N and started at position k We want to sho... View the full answer