Let X and Y be jointly distributed random variables This exercise leads you through a proof of the fact that 1 X,Y 1 a Express the quantity V(X (X Y) Y) in terms of X, Y, and Cov(X, Y ) b Use the fact that V(X (X Y)Y) 0 and Cov(X, Y) X,YXY to show that X,Y 1 c Repeat parts (a) and (b) using V(X (X ...

Let X and Y be jointly distributed random variables. This exercise leads you through a proof of

Question:

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) ≥ 0 and 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.