> [!NOTE] Definition > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a non-negative [[Discrete random variables|discrete real-valued random]] variable on $(\Omega,\mathcal{F},\mathbb{P}).$ Let $A\in\mathcal{F}$ so that $\mathbb{P}(B)>0.$ Then the conditional expectation of $X$ over $A$ is given by $\mathbb{E}[X\mid A]=\sum_{x\in D_{X}} x \cdot \mathbb{P}(X^{-1}(x)\mid A)$where $\mathbb{P}(B\mid A)$ denotes the [[Conditional Probability|conditional probability]] of $B$ given $A$ and $D_{X}$ denotes the [[Discrete Support of Distribution of Discrete Real-Valued Random Variable|discrete support]] of $X.$ > # Applications See [[Conditional Expectation of Discrete Real-Valued Random Variable Over Another]]. By [[Expectation of Conditional Expectation of Discrete Random Variable Over Another Equals Expectation (Total Expectation Theorem)]], ....