> [!NOTE] Theorem > Let $X, Y$ be [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable discrete real-valued random variables]]. Then the [[Covariance of Square-Integrable Discrete Real-Valued Random Variables|covariance]] of $X$ with itself is the [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] of $X$: that is, $\text{Cov}(X,X)= \text{Var}(X).$ **Proof**: We have $\begin{align} \text{Cov}(X,X) &= \mathbb{E}[(X-\mathbb{E}[X])^{2}] \\ &= \text{ Var}(X) \end{align}$