Explanation:

Statement D is false because correlation measures only linear relationships between variables. A correlation of zero indicates no linear relationship, but there could still be non-linear dependencies between the variables.

Key points:

• Correlation (ρ) = Cov(X,Y) / (σ_X × σ_Y)
• Correlation = 0 means no linear relationship
• Dependence ≠ 0 could still exist through non-linear relationships
• Independence implies correlation = 0, but correlation = 0 does NOT imply independence

Examples of zero correlation but non-zero dependence:

1. X ~ Uniform(-1,1), Y = X² → Correlation = 0 but clear quadratic relationship
2. Circular relationships where points form a circle pattern
3. Other non-linear functional relationships

Correct statements:

• A: True - Both measure linear relationship strength
• B: True - This is the mathematical definition of correlation
• C: True - This is the definition of statistical independence