FRM Part I · FRM Exam Part I · Random Variables
X takes the values −1, 0 and +1 with equal probability, and Y = X². Which statement is correct?
Covariance is zero, yet X and Y are not independent. E[X] is 0 and E[XY] = E[X³] is 0, so the covariance vanishes. However, Y is completely determined by X, so a nonlinear dependence exists that covariance cannot detect.
- ACov(X,Y) = 0, yet X and Y are not independentCorrect
- BCov(X,Y) > 0 because Y increases when |X| increases
- CCov(X,Y) = 0, so X and Y are independent
- DCov(X,Y) < 0 because Y is never negative while X can be
Explanation
E[X] = 0 and E[XY] = E[X³] = (−1 + 0 + 1)/3 = 0, so Cov = 0 − 0 × E[Y] = 0. But Y is fully determined by X (for example, P(Y=1 | X=1) = 1 while P(Y=1) = 2/3), so they are dependent. Zero covariance captures only linear dependence.
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