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IAI Actuarial Core Principles · Actuarial Statistics · Expectations and conditional expectations

X and Y are random variables with Var(X) = 16, Var(Y) = 25 and correlation coefficient −0.5. What is Var(3X − 2Y)?

Var(3X − 2Y) is 364. With covariance −10, the variance is 144 + 100 − 12×(−10) = 364, since the negative covariance adds to the variance when the coefficients have opposite signs.

  1. A244
  2. B304Correct
  3. C144
  4. D364
  5. 424

Explanation

Cov(X,Y) = −0.5 × 4 × 5 = −10. Var(3X − 2Y) = 9×16 + 4×25 − 2×3×2×Cov = 144 + 100 − 12(−10) = 144 + 100 + 120 = 364. The key is therefore 364, at index 3. Using +12×Cov would give 124; ignoring covariance gives 244.

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