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FRM Part I · FRM Exam Part I · Fundamentals of Probability

X is equally likely to be -1, 0 or +1, and Y = X². Which statement is correct?

Covariance is zero but the variables are dependent. E[X] and E[X³] are both zero, so Cov(X, X²) = 0. However, Y is completely determined by X, so they are not independent. Correlation measures only linear dependence and misses this nonlinear relationship.

  1. ACov(X,Y) = 0, yet X and Y are not independentCorrect
  2. BCov(X,Y) = 0, so X and Y are independent
  3. CCov(X,Y) > 0 because Y is a function of X
  4. DCorr(X,Y) = 1 because Y is perfectly determined by X

Explanation

E[X] = 0 and E[XY] = E[X³] = 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 does not imply independence, and correlation only captures linear dependence.

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