Chapter 4.1, Problem 9E

A particle starts at $\left(0,0\right)$ and moves in one-unit independent steps with equal probabilities of $\frac{1}{4}$ in each of the four directions: north, south, east, and west. Let S equal the east—west position and T the north-south position after ii steps.

(a) Define the joint pmf of S and T with $n=3$.

On a two dimensional graph, give the probabilities of the joint pmf and the marginal pmfs (similar to Figure 4.1-1).

(b) Let $X=S+3$ and let $Y=T+3$.

Give the marginal pmfs of X and Y.