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

Which statement about expectations of random variables X and Y is correct in general?

The expected value of a sum equals the sum of expected values, whatever the dependence between the variables. Product expectation and additive variance require independence or zero covariance, and E[X squared] equals the squared mean only when variance is zero.

  1. AE[XY] = E[X]E[Y] for any X and Y
  2. BE[X + Y] = E[X] + E[Y] regardless of dependenceCorrect
  3. CVar(X + Y) = Var(X) + Var(Y) for any X and Y
  4. DE[X^2] = (E[X])^2 for any X

Explanation

Expectation is linear, so the expectation of a sum equals the sum of expectations whether or not X and Y are independent. The product and variance-additivity rules need independence (or zero covariance), and E[X^2] exceeds (E[X])^2 unless variance is zero.

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