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

Which statement about covariance and independence of two random variables X and Y is correct?

Independent variables always have zero covariance, because independence implies the expectation of the product equals the product of expectations. Zero covariance does not imply independence, since nonlinear dependence can exist, so the reverse statement is false.

  1. AIf Cov(X,Y) = 0, X and Y must be independent
  2. BIf X and Y are independent, Cov(X,Y) = 0Correct
  3. CIf X and Y are independent, their correlation equals 1
  4. DIf Cov(X,Y) = 0, then Var(X+Y) = Var(X) − Var(Y)

Explanation

Independence implies E[XY] = E[X]E[Y], so covariance is zero. The converse fails because covariance only captures linear dependence (e.g., Y = X² with symmetric X). With zero covariance, Var(X+Y) = Var(X) + Var(Y), not the difference.

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