## Prove that Cov(aX, bY) = ab Cov(X,Y).

1.  Let X and Y be random variables, and a and b be constants.

a.  Prove that Cov(aX, bY) = ab Cov(X,Y).

b.  Prove that if a > 0and b > 0, then ρaX,bY = ρX,Y .

c.   Conclude that the correlation coefficient is unaffected by changes in units.

2.   Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z). (Hint: Use Equation 2.71.)

3.  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.

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