We will now examine an elementary example of a random walk. In Chapter 12, we will revisit this concept and its applications.

Suppose that a particle is at 0 on the integer number line and suppose that at step 1, the particle will move to 1 with probability p, 0 < p < 1, and will move to −1 with probability 1 − p.

Furthermore, if at step n, the particle is at i, then independently of the previous moves, it will move 1 unit to the right to i + 1 with probability p and will move 1 unit to the left to i − 1 with probability 1−p. Let X be the position of the particle after n moves. Find the probability mass function of X.