> [!NOTE] Definition 1 (Integrable of Discrete Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Discrete random variables|discrete real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Then $X$ is integrable iff its [[Expectation of Discrete Real-Valued Random Variable|expectation]] is defined: that is the sum $\sum_{x:\mathbb{P}(X=x)>0}|x|\cdot \mathbb{P}(X=x)$[[Convergent Real Series|converges]]. **Notation**: $X=x$ denotes the set $\{ \omega: X(\omega) = x\}$ or $X^{-1}(x),$ the [[Preimage (of set under a function)|preimage]] of $x$ under $X.$ # Properties