**Theorem** If $X$ and $Y$ are [[Independence of Two Discrete Real-Valued Random Variables]] that are [[Integrable Discrete Real-Valued Random Variable|integrable]], then $XY$ is integrable and $\mathbb{E}[XY]=\mathbb{E}[X] \cdot \mathbb{E}[Y].$ **Proof** Using [[Expectation of Real-Valued Function of Bivariate Discrete Real-Valued Random Variable]], $\begin{align} \mathbb{E}[XY] &= \sum_{X \in D_{X}} \sum_{y \in D_{Y}} xy \cdot \mathbb{P}(X=x,Y=y) \\ &= \sum_{x \in D_{X}} x \cdot \left( \sum_{y \in D_{Y}} y \cdot \mathbb{P}(X=x)\mathbb{P}(Y=y) \right) \\ &= \sum_{x \in D_{X}} x \cdot \mathbb{P}(X=x) \cdot \left( \sum_{y \in D_{Y}} y \cdot \mathbb{P}(Y=y)\right) \\ &= \mathbb{E}[Y] \cdot \mathbb{E}[X] \end{align}$