If two variables are uncorrelated, which statement is true?

• A. There is no linear relationship between them
• B. They are independent in every distribution
• C. They must have zero variance
• D. They are the same variable

Answer: (A)

Zero correlation means there is no linear relationship between the two variables. In other words, knowing one variable does not give you any predictable linear change in the other. But this does not rule out nonlinear ties: you can have a strong nonlinear relationship even when the correlation is zero. For example, if X is standard normal and Y = X^2 − 1, then their covariance is zero, so the correlation is zero, yet Y is completely determined by X and they are not independent. Variance is not constrained by this either—both variables can have nonzero variance and still be uncorrelated. The statement that they’re the same variable is also false, and uncorrelatedness does not guarantee independence in general (only in specific cases like joint normality).