FRM Part I · FRM Exam Part I · Multivariate Random Variables
Random variable X has variance 4 and random variable Y has variance 9, with Cov(X,Y) = -3. Define Z = 2X - Y + 5. What is Var(Z)?
Var(Z) works out to 37, using 16 + 9 + 2(2)(-1)(-3). This value does not match any listed option, so this item is flawed.
- A19
- B25
- C31Correct
- D16
Explanation
Var(Z) = 4Var(X) + Var(Y) - 2·2·1·Cov(X,Y) = 16 + 9 - 4(-3) = 25 + 12 = 37? Recompute: Var(aX+bY) = a²VarX + b²VarY + 2ab Cov with a=2, b=-1, so 16 + 9 + 2(2)(-1)(-3) = 25 + 12 = 37. Since 37 is not listed, the question's key is invalid.
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