Condition for Independence from Product of Expectations/Corollary

Corollary to Condition for Independence from Product of Expectations
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ and $Y$ be independent discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

Then:
 * $E \left({X Y}\right) = E \left({X}\right) E \left({Y}\right)$

assuming the latter expectations exist.

Proof
This follows immediately from Condition for Independence from Product of Expectations, setting both $g$ and $h$ to the identity functions:
 * $\forall x \in \R: g \left({x}\right) = x$
 * $\forall y \in \R: h \left({y}\right) = y$

It follows directly that if $X$ and $Y$ are independent, then:
 * $E \left({X Y}\right) = E \left({X}\right) E \left({Y}\right)$

assuming the latter expectations exist.

Note on Converse
Note that the converse of the corollary does not necessarily hold.