> [!NOTE] Definition 1 (Square-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 square-integrable iff [[Expectation of Discrete Real-Valued Random Variable|expectation]] of $X^{2}$ defined: that is the sum $\sum_{x:\mathbb{P}(X=x)>0}x^{2}\cdot \mathbb{P}(X=x)$is [[Convergent Real Series|convergent]]. # Properties By [[Square-Integrable Discrete Real-Valued Random Variable is Integrable]], if a random $X$ is square-integrable, then it is integrable. The [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] of a square integrable discrete real-valued random variable $X$ is defined as the expectation of the squares of the deviations of the values of $X$ from the expectation of $X.$