Predicting Extraction for an Existing Tower with a Given Number of Steps. An existing tower contains 5.0 theoretical steps. It is desired to predict its performance under the following conditions….
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.