Answer: -3.2

## 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)

Author: Tanishq Prabhu