Question

4. Consider a random walk on the integers in which the particle moves either two units to the right (with probability p) or one unit to the left (with probability q1 p) at each stage where O < p1. There is an absorbing barrier at 0 and the particle starts at the point a (> 0). Show that the probability π (a) that the particle is ultimately absorbed at 0 satisfies the difference equation and deduce that π (a) l if p < ģ Suppose that the particle is absorbed whenever it hits either N or N + 1. Find the probability T N (a) that it is absorbed at O rather than at N or N +1, having started at a, where 0 a < N I. Deduce that, as N → oo, 1 if p 3 N (a) Deduce that if a fair coin is tossed repeatedly, the probability that the number of heads ever exceeds twice the number of tails is 5 1).

I don't know if this one question counts as multiple questions. If it does, please help me with the first part and tell me that it counts as multiple parts.

Answer 1