Consider a walker who walks on a real line starting at 0 with moving one step forward with probability p and backward with probability q, p q + = 1

Consider a walker who walks on a real line starting at 0 with moving one step forward with probability p and backward with probability q, p q + = 1. Let Xn describe the position of the walker after n steps.

a. What is the probability that walker is at the point 0 on the line after two steps?

b. What is the probability that walker is at the point −1 on the line after three steps?

c. What is the probability that walker is at the point 3 on the line after three steps?

d. Suppose the walker is at point 4 after 10 steps, does the probability that it will be at point 8 after 16 steps (6 more steps) depend on how it moves to point 4 within the first 10 steps?

e. Are X10 − X4 and X16 − X12 independent?

f. Are X10 − X4 and X12 − X8 independent?