IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
X and Y have Var(X) = 4, Var(Y) = 25 and Cov(X, Y) = 6. Let U = X + Y and V = X - Y. What is Cov(U, V)?
The covariance is -21. Expanding Cov(X+Y, X-Y) gives Var(X) minus Var(Y), because the cross covariance terms cancel. That is 4 minus 25, which is -21. Reversing the order would wrongly give +21.
- A-21Correct
- B21
- C10
- D-10
- 31
Explanation
Cov(X+Y, X-Y) = Var(X) - Cov(X,Y) + Cov(Y,X) - Var(Y) = Var(X) - Var(Y) = 4 - 25 = -21. The covariance terms cancel, so the 6 is irrelevant. Option 21 reverses the sign.
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