5. Consider a simple random walk with an absorbing barrier at 0 and a ‘retaining’ barrier at N.

That is to say, the walk is not allowed to pass to the right of N, so that its position Sn at time n satisfies P(Sn+1 = N | Sn = N) = p, P(Sn+1 = N − 1 | Sn = N) = q, where p +q = 1. Set up a difference equation for the mean number e

(a) of jumps of the walk until absorption at 0, starting from

a, where 0 ≤ a ≤ N. Deduce that e

(a) = a(2N − a + 1) if p = q = 1 2 , and find e

(a) if p 6= q.