If we have two r.v.: X = { -1, 0, 1 } and Y= , with X uniformly distributed with probability 1/3, then X and Y are independent, right?

Pdf values of Y are 2/3 if Y=1 and 1/3 if Y=0.

I can construct joint pdf with f(x, y), such that f(i, -1) = 0, f(i, 0) = 1/9 and f(i, 1) = 2/9. Then for every f(i, j), f(x, j) = f(x)f(y), and that is the definition of independence.

Am I missing something?