> [!NOTE] Theorem > Let $X,Y$ be [[Independence of Two Discrete Real-Valued Random Variables|independent discrete real-valued random variables]] that are also [[Integrable Discrete Real-Valued Random Variable|integrable]]. Then $XY$ is also integrable and its [[Expectation of Discrete Real-Valued Random Variable|expectation]] is given by $\mathbb{E}[XY]=\mathbb{E}[X]\cdot \mathbb{E}[Y]$ **Proof**: By [[Expectation of Real-Valued Function of Bivariate Discrete Real-Valued Random Variable]], $\begin{align} \mathbb{E}[XY]&=\sum_{x\in D_{X}} \sum_{y\in D_{Y}} xy \cdot \mathbb{P}(X=x,Y=y) \\ &= \sum_{x\in D_{X}} x \left( \sum_{y\in D_{Y}} y\cdot \mathbb{P}(X=x)\mathbb{P}(Y=y) \right) \\ &= \sum_{x\in D_{X}} x \cdot \mathbb{P}(x=x) \left( \sum_{y\in D_{Y}} y\cdot \mathbb{P}(Y=y) \right) \\ &= \left( \sum_{y\in D_{Y}} y\cdot \mathbb{P}(Y=y) \right) \left( \sum_{x\in D_{X}} x \cdot \mathbb{P}(x=x) \right) \\ &= \mathbb{E}[X]\cdot \mathbb{E}[Y] \end{align}.$