FRM Part I · FRM Exam Part I · Sample Moments
X takes the values -1, 0 and 1 with equal probability, and Y = X². Which statement about the covariance and correlation of X and Y is correct?
Covariance and correlation are both zero even though Y is fully determined by X. The expected value of X is zero and E[XY] equals E[X cubed], which is also zero. Correlation measures only linear dependence, so zero correlation does not imply independence.
- ACovariance and correlation are both zero even though Y is completely determined by XCorrect
- BCorrelation is +1 because Y is an exact function of X
- CCovariance is zero, which proves X and Y are independent
- DCorrelation is negative because Y is zero when X is zero
Explanation
E[X] = 0, E[Y] = 2/3, and E[XY] = E[X³] = 0, so Cov = 0 - 0 = 0 and the correlation is 0. Correlation captures only linear dependence, and the relationship here is perfectly nonlinear. Zero covariance therefore does not imply independence, which rules out the third option.
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