> [!NOTE] Definition 1 (Standard Deviation of Square-Integrable of Discrete Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable discrete real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Then the standard deviation of $X$ is given by $\sigma(X)=\sqrt{ \text{Var}(X) },$where $\text{Var}(X)$ denotes the [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] of $X.$ # Properties By [[Variance of Linear Transformation of Square-Integrable Discrete Real-Valued Random Variable]], for all $a\in \mathbb{R},$ $\sigma(aX+b)=|a|\cdot\sigma(x).$