> [!NOTE] Theorem > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $X:\Omega\to \{ 0,1,2,\dots \}$ be a non-negative [[Discrete random variables|discrete real-valued random]] variable on $(\Omega,\mathcal{F},\mathbb{P}).$ Then its [[Expectation of Discrete Real-Valued Random Variable|expectation]] satisfies $\mathbb{E}[X] = \sum_{i=0}^{\infty}\mathbb{P}(X>i)$ **Proof**: By [[Law of Total Probability]], $\sum_{i\geq 0}\mathbb{P}(X>i)=\sum_{i\geq 0}\sum_{j \geq i+1} \mathbb{P}(X=j)=\sum_{i \geq 0} i \cdot \mathbb{P}(X=i) = \mathbb{E}[X] $