Two dependent random variables with $E[XY]=E[X]E[Y]$ while $E[X] \not = 0$ and $E[Y] \not = 0$.

Question

Please, help me find two dependent random variables with $E[X] \not = 0$, $E[Y] \not = 0$ and $E[XY] = E[X]E[Y]$.

Answer 1

\begin{align} X & = \begin{cases} 10 \\ 11 & \text{each with probability } 1/3. \\ 12 \end{cases} \\[12pt] Y & = (X-11)^2 \end{align}