FRM Part I · FRM Exam Part I · Multivariate Random Variables
Which statement about independence and correlation of two random variables is correct?
Independence implies zero covariance when the moments exist, because the expected value of the product equals the product of the expected values. The reverse is not true in general, since uncorrelated variables can still be nonlinearly dependent.
- AZero correlation implies independence for any joint distribution
- BIndependence implies zero covariance, provided the moments existCorrect
- CIndependence implies that the conditional variance of Y given X must be zero
- DNonzero covariance is compatible with independence if variances differ
Explanation
If X and Y are independent, E[XY]=E[X]E[Y], so covariance is zero when moments exist. The converse fails in general, e.g. Y=X² with symmetric X. Conditional variance under independence equals the unconditional variance, not zero.
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