Answer: 3V[Y]

**Step-by-step solution:** 1. **Given relationships:** - V[X] = 4V[Y] - Cov[X,Y] = V[Y] 2. **Define E and F:** - E = X + Y - F = X - Y 3. **Calculate Cov[E, F]:** Cov[E, F] = Cov[X + Y, X - Y] 4. **Expand using covariance properties:** Cov[X + Y, X - Y] = Cov[X, X] - Cov[X, Y] + Cov[Y, X] - Cov[Y, Y] 5. **Simplify using covariance properties:** - Cov[X, X] = V[X] - Cov[Y, X] = Cov[X, Y] (covariance is symmetric) - Cov[Y, Y] = V[Y] So: Cov[E, F] = V[X] - Cov[X,Y] + Cov[X,Y] - V[Y] 6. **Cancel terms:** Cov[E, F] = V[X] - V[Y] 7. **Substitute given relationship V[X] = 4V[Y]:** Cov[E, F] = 4V[Y] - V[Y] = 3V[Y] **Key concepts used:** - Covariance properties: Cov(aX + bY, cW + dZ) = acCov(X,W) + adCov(X,Z) + bcCov(Y,W) + bdCov(Y,Z) - Cov(X,X) = V[X] - Cov(X,Y) = Cov(Y,X) (symmetry) - Variance is a special case of covariance where both variables are the same

Author: Nikitesh Somanthe