> [!NOTE] Definition 1 (Variance 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 variance of $X$ is given by $\text{Var}(X)=\mathbb{E}[(X-\mu)^{2}],$where $\mu=\mathbb{E}[X]$: the [[Expectation of Discrete Real-Valued Random Variable|expectation]] of $X.$ > [!NOTE] Definition 2 (Variance 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 variance of $X$ is given by $\text{Var}(X)=\mathbb{E}[X^{2}]-(\mathbb{E}[X])^{2},$where $\mathbb{E}[X]$ denotes the [[Expectation of Discrete Real-Valued Random Variable|expectation]] of $X.$ **Note**: By [[Equivalence of Definitions of Variance of a Square-Integrable Discrete Real-Valued Random Variable]], these definitions are indeed equivalent. The second is far easier to work with because in practice, we can use [[Expectation of Real-Valued Function of Discrete Real-Valued Random Variable]] to work out $\mathbb{E}[X^{2}]$. # Properties By [[Variance of Linear Transformation of Square-Integrable Discrete Real-Valued Random Variable]], for all $a\in \mathbb{R},$ $\text{Var}(aX+b)=a^{2}\text{Var}(X).$ By [[Covariance of Square Integrable Discrete Real-Valued Random Variable with Itself]], $\text{Var}(X)=\text{Cov}(X,X).$ # Applications **Examples**: By [[Variance of Poisson Distribution]], if $X\sim \text{Poisson}(\lambda)$ then $\text{Var}(X)=\lambda.$ By [[Variance of Geometric Distribution]], if $X\sim \text{Geom}(p),$ then $\text{Var}(X)=(1-p)/p^{2}.$ By [[Variance of Bernoulli Distribution]], if $X \sim \text{Bernoulli}(p),$ then $\text{Var}(X)=p\cdot( 1-p).$ **Standard deviation**: Note that $\text{Var}(aX)=a^{2}\cdot \text{Var}(X)$ which means that $\text{Var}(X)$ does not have the same unit as $X.$ Thus we define [[Standard Deviation of Square-Integrable Discrete Real-Valued Random Variable|standard deviation]] as the square root of its variance so that it has the same unit.