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).