Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that -1 ≤ ρX,Y ≤ 1.

a. Express the quantity V(X − (σX/σY)Y) in terms of σX, σY, and Cov(X, Y).

b. Use the fact that V(X − (σX/σY)Y) ≥ 0and Cov(X, Y) = ρX,YσXσY to show that ρX,Y ≤ 1.

c. Repeat parts (a) and (b) using V(X + (σX/σY)Y) to show that ρX,Y ≥ −1.

Step 1 of 4:

It is given that X and Y are two random variables and are jointly distributed.

Also it is known that -11.

Using these we have to prove the others.

Step 2 of 4:

(a)

Here we have to express V(X-()Y) in terms of ,,Cov(X,Y).

We know that

V(X)=E()-

Thus,

V(X-()Y)=E-

=E(-2XY)-

=E()+E()-E(2XY)-+-2E(X)E()]

=[E()-]-2[E(XY)-E(x)E(Y)]+[E()-]

=-2Cov(X,Y)+

=2[-Cov(X,Y)]

Hence,

V(X-()Y)=2[-Cov(X,Y)]