Answer to Question #212757 in Statistics and Probability for Tshimo

Question

2.  Show that

  (a) var(X – Y) = var(X) + var(Y) – 2cov(X,Y)                                                                                  (b) cov(X,aY + b) = acov(X,Y) if a and are constants.         

Answer 1

(a)

$$Var(X-Y)=E[(X-Y)^2]-(E[X-Y])^2$$

$$=E[X^2-2XY+Y^2]-(\mu_X-\mu_Y)^2$$

$$=E[X^2]-2E[XY]+E[Y^2]-\mu_X^2+2\mu_X\mu_Y-\mu_Y^2$$

$$=(E[X^2]-\mu_X^2)+(E[Y^2]-\mu_Y^2)-2(E[XY]-\mu_X\mu_Y)$$

$$=Var(X)+Var(Y)-2Cov(X, Y)$$

(b)

$$Cov(X, aY+b)=E((X-E(X))(aY+b-E(aY+b)))$$

$$=E((X-E(X))(aY+b-aE(Y)-b))$$

$$=E((X-E(X))(aY-aE(Y)))$$

$$=aE((X-E(X))(Y-E(Y)))$$

$$=aCov(X, Y),$$

$a$ and $b$ are constants.