Variance of a sum of variables
Task number: 3551
Show that for independent random variables \(X\) and \(Y\) it is true that \(var(X+Y)=var(X)+var(Y)\).
Hint
Use the fact that for independent random variables \(E[XY]=EXEY\).
Solution
\(var(X+Y)=\\ E[(X+Y)^2]-(E[X+Y])^2=\\ E[X^2+2XY+Y^2]-(EX+EY)^2=\\ E[X^2]+2E[XY]+E[Y^2]-((EX)^2+2EXEY+(EY)^2)=\\ E[X^2]-(EX)^2+E[Y^2]-(EY)^2+2(E[XY]-EXEY)=\\ var(X)+var(Y)\).
