Is it possible given random variables X and Y for $E[XY] = E[X]E[Y]$ if X and Y are dependent variables?

Question

Given random dependent variables X and Y, can we choose some X and Y such that $E[XY] = E[X]E[Y]$?

Answer 1

Take $X$ to be any symmetric random variable about 0, and $Y=X^2$. Then $E[XY]=E[X^3]=0=E[X]E[Y]=0\cdot E[X^2]$.