If $EX^2 < \infty$ and $EY^2 < \infty$, are we guaranteed that $EXY < \infty$ I was wondering under what conditions does $EX^2 < \infty$ and $EY^2 < \infty$, guarantee that $EXY < \infty$? Or is this just false usually? 
I tried to come up with some counterexamples, but had little success.

Hint:
$$|x-y|^2\ge 0$$
so
$$x^2+y^2\ge 2|xy|$$
