Discrete example of uncorrelated but not independent random variables

Question

The simplest example I could think of regarding the title is to let $X$ be uniform on $\mathbb{Z}\cap[-2,2]$ and $Y=X^2$. However, all the examples I see are slightly more complicated. Does my example indeed work? I want to check I'm not missing something.

Answer 1

A slightly easier example is with $X$ uniform on $\{-1,0,1\}$ and $Y=X^2$. They are not independent because $P(Y=0)=1/3$ while $P(Y=0|X=0)=1$. They are uncorrelated because $EX=0$ and $EXY=EX^3=0$, and hence $\text{COV}(X,Y)=EXY-EXEY=0-0=0$.