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FRM Part I · FRM Exam Part I · Multivariate Random Variables

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

If X and Y are independent, their covariance is zero, provided it exists, because the expectation of the product equals the product of expectations. The reverse is false, since zero covariance only rules out linear dependence and nonlinear dependence such as Y equal to X squared can remain.

  1. AIf Cov(X,Y)=0, then X and Y must be independent
  2. BIf X and Y are independent, then Cov(X,Y)=0, provided the covariance existsCorrect
  3. CIf X and Y are independent, then Var(X+Y) exceeds Var(X)+Var(Y)
  4. DIf E[Y|X] varies with X, then Cov(X,Y) must equal zero

Explanation

Independence implies E[XY]=E[X]E[Y], so covariance is zero when it exists. The converse fails: for example, Y=X^2 with X symmetric about zero has zero covariance but is dependent. For independent variables, variance of the sum equals the sum of variances.

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