> [!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 $(\Omega, \mathcal{F}, \mathbb{P})$ such that for all $\omega \in \Omega,$ $X(\omega)\leq Y(\omega).$ Then $\mathbb{E}[X]\leq \mathbb{E}[Y],$ where $\mathbb{E}[X]$ denotes the [[Expectation of Discrete Real-Valued Random Variable|expectation]] of $X.$ **Proof**: **Proof**: Follows from [[Expectation is Linear]].