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.