Condition for Independence from Product of Expectations/Corollary/Converse

Converse of Corollary to Condition for Independence from Product of Expectations
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$ such that:
 * $\expect {X Y} = \expect X \expect Y$

Then it is not necessarily the case that $X$ and $Y$ are independent.

Proof
Proof by Counterexample:

Let $X$ be a discrete random variable whose distribution is defined as:
 * $\map {p_X} {-1} = \map {p_X} 0 = \map {p_X} 1 = \dfrac 1 3$

Let $Y$ be the discrete random variable defined as:
 * $Y = \begin{cases}

0 & : X = 0 \\ 1 & : X \ne 0 \end{cases}$

We have:

So $X$ and $Y$ are dependent.

But:

So $\expect {X, Y} = \expect X \expect Y$ but $X$ and $Y$ are not independent.