Probability Practice Questions

Question

Show that if X and Y are independent random variables, then Var(X+Y)= Var(X)+Var(Y).

Answer 1

var(X+Y) = E(X+Y-E(X+Y))^2 = E(X-E(X)+Y-E(Y))^2 = E(X-E(X))^2+E(Y-E(Y))^2 +2E((X-E(X))(Y-E(Y)))=

E(X-E(X))^2+E(Y-E(Y))^2 +2E(XY -XE(Y)-E(X)Y+E(X)E(Y)) =

E(X-E(X))^2+E(Y-E(Y))^2 +2E(XY) -2E(XE(Y))-2E(E(X)Y)+2E(E(X)E(Y)) =

E(X-E(X))^2+E(Y-E(Y))^2 +2E(XY) -2E(X)E(Y)-2E(X)E(Y)+2E(X)E(Y) =

E(X-E(X))^2+E(Y-E(Y))^2 + 2E(XY) - 2E(X)E(Y)

This is the point at which you use independence, which tells us that E(XY)=E(X)E(Y)

Thus,

E(X-E(X))^2+E(Y-E(Y))^2 + 2E(XY) - 2E(X)E(Y) =

E(X-E(X))^2+E(Y-E(Y))^2 + 2E(X)E(Y) - 2E(X)E(Y) =

E(X-E(X))^2+E(Y-E(Y))^2 =

var(X) + var(Y)