1. In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective,….

## 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.