# Sum of Random Variables

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:

$$Cov(X,Y)= 0$$ $$Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)$$

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