Condition for Independence from Product of Expectations/Corollary

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

Then:
 * $\expect {X Y} = \expect X \expect Y$

assuming the latter expectations exist.

Proof
From Condition for Independence from Product of Expectations, setting both $g$ and $h$ to the identity functions:
 * $\forall x \in \R: \map g x = x$
 * $\forall y \in \R: \map h y = y$

It follows directly that if $X$ and $Y$ are independent, then:
 * $\expect {X Y} = \expect X \expect Y$

assuming the latter expectations exist.

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