If X and Y are random variables, sometimes we need to know the mean
and variance of their sum, X + Y, or weighted sum, aX + bY. For instance, total
revenue = price1 × vol1+ price2 × vol2.
$$E(aX + bY)=aE(X)+bE(Y)=aμ_X+bμ_Y$$
$$Var(aX+bY)=a^2 Var(X) + b^2 Var(Y) + 2abCov(X,Y)$$
$$Var(aX+bY)=a^2 Var(X) + b^2 Var(Y) + 2abσ_X σ_Y Corr(X,Y)$$
If X and Y are independent:
$$Cov(X,Y)= 0$$
$$Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)$$
