Definition

Let $X$ and $Y$ be RVs, their Covariance is defined as $$Cov(X,Y)=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]$$ # Proposition (Properties of Covariance) 1. Linearity: For $\alpha_1,\alpha_2\in\mathbb{R}$ $$Cov(X,\alpha_1Y_1+\alpha_2Y_2)=\alpha_1Cov(X,Y_1)+\alpha_2Cov(X,Y_2)$$ 2. Independence: If $X$ and $Y$ are independent then $$Cov(X,Y)=0$$ # Lemma (Variance Identity) $$Var(\alpha X+\beta Y)=\alpha^2Cov(X,X)+\beta^2Cov(Y,Y)+2\alpha\beta Cov(X,Y)$$