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