A particle starts at (0, 0) and moves in one independent steps with equal probabilities of 1 /4 in each of the four directions: north, south, east, and west. Let (X, Y ) be the location of the particle after n steps.

Describe the probability mass function of X and Y .

That is, for all integer k, compute P(X = k) and P(Y = k).

(There is no need to simplify.)