FRM Part I · FRM Exam Part I · Multivariate Random Variables
Random variable X has variance 4 and random variable Y has variance 9. Their correlation is -0.5. What is the variance of Z = 2X - Y?
The variance of Z is 37. Covariance is -0.5 times 2 times 3, which is -3. For 2X minus Y, variance is 4 times 4 plus 9 minus 2 times 2 times (-3), giving 16 + 9 + 12 = 37. The negative correlation raises the variance of a difference.
- A37Correct
- B25
- C13
- D19
Explanation
Cov(X,Y) = -0.5 × 2 × 3 = -3. Var(Z) = 4(4) + 9 - 2(2)(1)(-3) = 16 + 9 + 12 = 37. Using a plus sign on the covariance term gives 13 (16+9-12), and ignoring covariance gives 25.
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