> [!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 $X$ and $Y$ satisfies $\text{Cov}(X,Y)= \text{Cov}(Y,X).$ **Proof**: We have $\begin{align} \text{Cov}(X,Y) & = \mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])] \\ &= \mathbb{E}[(Y-\mathbb{E}[Y])(X-\mathbb{E}[X])] \\ &= \text{Cov}(Y,X). \end{align}$