Step-by-step solution:

1. Given values:

   • Correlation between X and Y, ρ(X,Y) = 0.50
   • Standard deviation of Y, σᵧ = 0.26
   • Covariance between X and Y, Cov(X,Y) = 0.005
   • We need to find variance of X, Var(X) = σₓ²

2. Correlation formula: ρ(X,Y) = Cov(X,Y) / (σₓ × σᵧ)

3. Substitute known values: 0.50 = 0.005 / (σₓ × 0.26)

4. Solve for σₓ: σₓ × 0.26 = 0.005 / 0.50 σₓ × 0.26 = 0.01 σₓ = 0.01 / 0.26 σₓ = 0.0384615 ≈ 0.0385

5. Calculate variance of X: Var(X) = σₓ² = (0.0385)² = 0.00148225 ≈ 0.00148

Key points:

• Correlation is the standardized covariance
• Variance is the square of standard deviation
• The calculation requires solving for the unknown standard deviation first, then squaring it to get variance
• Option C (0.0385) is the standard deviation, not the variance
• Option D (0.0148) is off by a factor of 10
• Option A (0.13) is incorrect