uncorrelated, not independent with given expectations

Question

I want to devise two discrete random variables X and Y with the given expectations $E(X)=10$ and $E(Y)=20$, that are uncorellated ($E(XY)=200$), but dependent. Is there an easy distribution for that? I'm trying to tweak the classic $P(X=-1, Y=0)=P(X=1,Y=0)=P(X=0,Y=-1)=P(X=0,Y=1)=0.25$ and keep getting it wrong...

Simple examples of uncorrelated but not independent $X$ and $Y$

Answer 1

What happens if you try $X^\prime=X+10$ and $Y^\prime=Y+20$ with your classic example?

so

$$\begin{matrix} & P(X^\prime=9,Y^\prime=20)\\= & P(X^\prime=11,Y^\prime=20)\\= & P(X^\prime=10,Y^\prime=19)\\= & P(X^\prime=10, Y^\prime=21)\\= & 0.25\end{matrix}$$