Answer: -3.2

## Explanation We need to find Cov(Z, V) where V = 3X - 2Y. ### Step 1: Apply covariance properties Using the linearity property of covariance: Cov(Z, V) = Cov(Z, 3X - 2Y) = 3Cov(Z, X) - 2Cov(Z, Y) ### Step 2: Use independence information We are told that X is independent of both Y and Z, so: Cov(Z, X) = 0 (since independence implies zero covariance) Thus: Cov(Z, V) = 3 × 0 - 2Cov(Z, Y) = -2Cov(Z, Y) ### Step 3: Calculate Cov(Z, Y) We know the correlation coefficient between Y and Z is ρ(Y,Z) = 0.8, and both have variance σ² = 2. Recall the relationship between covariance and correlation: ρ(Y,Z) = Cov(Y,Z) / (σ_Y × σ_Z) Since both have equal variance σ² = 2, their standard deviations are: σ_Y = σ_Z = √2 Therefore: Cov(Y,Z) = ρ(Y,Z) × σ_Y × σ_Z = 0.8 × √2 × √2 = 0.8 × 2 = 1.6 Note: Cov(Z,Y) = Cov(Y,Z) = 1.6 ### Step 4: Final calculation Cov(Z, V) = -2 × Cov(Z, Y) = -2 × 1.6 = -3.2 ### Step 5: Verification - X independent of Z → Cov(Z,X) = 0 ✓ - Correlation ρ(Y,Z) = 0.8 → Cov(Y,Z) = 0.8 × √2 × √2 = 1.6 ✓ - Linear combination: Cov(Z, 3X - 2Y) = 3Cov(Z,X) - 2Cov(Z,Y) = 0 - 2×1.6 = -3.2 ✓ Therefore, the correct answer is **-3.2** (Option D).

Author: Nikitesh Somanthe