> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X,Y$ be [[Integrable Discrete Real-Valued Random Variable|integrable discrete real-valued random variables]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Then for all $a,b\in \mathbb{R},$ $\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]$ where $\mathbb{E}[X]$ denotes the [[Expectation of Discrete Real-Valued Random Variable|expectation]] of $X.$ **Proof**: Follows directly from [[Expectation of Real-Valued Function of Bivariate Discrete Real-Valued Random Variable]]: $\begin{align} \mathbb{E}[aX+bY]&=\sum_{x\in D_{X}}\sum_{y\in D_{Y}}(ax+by)\cdot \mathbb{P}(X=x,Y=y) \\ &= a \sum_{x\in D_{X}} \sum_{y\in D_{Y}} x \cdot \mathbb{P}(X=x,Y=y) + b\sum_{x\in D_{X}} \sum_{y\in D_{Y}} y \cdot \mathbb{P}(X=x, Y=y ) \\ &= a \sum_{x\in D_{X}} x \cdot \sum_{y\in D_{Y}} \mathbb{P}(X=x,Y=y) + b\sum_{x\in D_{X}} y \cdot \sum_{y\in D_{Y}} \mathbb{P}(X=x, Y=y ) \\ &= a \sum_{x\in D_{X}} x \cdot \mathbb{P}(X=x) + b\sum_{x\in D_{X}} y \cdot \mathbb{P}(Y=y) \end{align}$since by [[Marginal Probability Mass Function of Discrete Real-Valued Random Variable]], $p_{X}(x) = \sum_{y\in D_{Y}} \mathbb{P}(X=x,Y=y).$ Thus $\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y].$