4. Given two random variables X and Y, what is the variance of X given variance[Y] = 100, variance [4X-3Y] = 2,700 and the correlation between X and Y is 0.5?

A. 56.3

B. 113.3

C. 159.9

D.225.0

Correct answer is D

Explanation:

A is incorrect. +3 was used instead of -3 when solving. variance [4X-3Y] = 16*Var[X] + 9*Var[Y] + 2*4*(+3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]. Solve for Var[X] = 56.3.

B is incorrect. (Var[X]^(1/2)* Var[Y]^(1/2) is missing from the equation when solving. Variance [4X-3Y] = 16*Var[X] + 9*Var[Y] + 2*4*(-3)*correlation[X,Y]. Solve for Var[X] = 113.3.

C is incorrect. The factor of 2 is missing from the equation. Variance [4X-3Y] = 16*Var[X] + 9*Var[Y] + 2*4*(-3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]. Solve for Var[X] = 159.9.

D is correct. Using the theorems on variance and covariance, variance [4X-3Y] = 16*Var[X] + 9*Var[Y] + 2*4*(-3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]. Solve for Var[X] = 225.0.

Reference:  Murray R. Spiegel, John Schiller, and R. Alu Srinivasan, Probability and Statistics, Schaum's  Outlines, 2nd ed. (New York: McGraw-Hill, 2000), Chapter 3.

Type of Question: Quantitative Analysis

谢谢楼主