Let (Sn)n be a simple random walk starting at 0, i.e. a positive move in each time-step and a probability p of a negative move. Compute the following probabilities:
1) P(Sn = 1)
2) P(Sn = 5, Sn-1 = 3)
3) P(Sn = 5, Sn-1 = 2)
4) P(Tn = 3) where To,1 = min{n: Sn, Sn-1}
Compute the following expectations:
1) E[Sn] for all n > 0.
2) E[Sn+1 | Sn] for all n > 0.