Are the final position coordinates (X, Y) independent in a simple 2D random walk?

Question

The situation I am trying to ask about is the following:

Starting at the origin, a single step is taken at random either up, down, left, or right. If n steps are taken, then the final position after these n movements is (X, Y). Are X and Y independent?

Answer 1

No. $\Pr[ X = 0 \mid Y = n] = 1$, for instance.