If X Y Z and W are random variables then show that a CovXYZ

If X, Y, Z, and W are random variables, then show that:

a) Cov(X+-Y,Z)= Cov(X,Z)+-Cov(Y,Z)

b)Cov(X+Y,Z+W) = Cov(X,Z) +Cov(X,W) + Cov(Y,Z)+Cov(Y,W)

c) Cov(X+Y,X-Y) = Var(X)-Var(Y)

Solution

a) Cov(X+Y,Z)

= Cov(X,Z)+Cov(Y,Z)

Similarlly

Cov(X-Y,Z)

= Cov(X,Z)-Cov(Y,Z)

b)

Cov(X+Y,Z+W)

= Cov(X,Z) +Cov(X,W) + Cov(Y,Z)+Cov(Y,W)

c) Cov(X+Y,X-Y)

= Cov(X,X)-Cov(X,Y)+Cov(Y,X)-Cov(Y,Y)

= Var(X) - Cov(X,Y)+Cov(X,Y) - Var(Y)

= Var(X) - Var(Y)

Hence Cov(X+Y,X-Y) = Var(X)-Var(Y)