Explanation

To find the covariance between Z and V where V = 3X - 2Y:

Step 1: Apply covariance properties

Using the linearity property of covariance: $\text{Cov}(Z, V) = \text{Cov}(Z, 3X - 2Y) = 3\text{Cov}(Z, X) - 2\text{Cov}(Z, Y)$

Step 2: Simplify using independence

Since X is independent of both Y and Z: $\text{Cov}(Z, X) = 0$ Therefore: $\text{Cov}(Z, V) = 0 - 2\text{Cov}(Z, Y) = -2\text{Cov}(Z, Y)$

Step 3: Calculate Cov(Z, Y)

Given that the correlation coefficient between Y and Z is ρ = 0.8, and all variables have equal variance σ² = 2: $\text{Cov}(Z, Y) = ρ_{ZY} × σ_Z × σ_Y = 0.8 × \sqrt{2} × \sqrt{2} = 0.8 × 2 = 1.6$

Step 4: Final calculation

$\text{Cov}(Z, V) = -2 × 1.6 = -3.2$

Key points:

• When variables are independent, their covariance is zero
• Correlation coefficient formula: ρ = Cov(X,Y)/(σ_Xσ_Y)
• Covariance is linear: Cov(aX + bY, Z) = aCov(X,Z) + bCov(Y,Z)