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

X and Y are standardized bivariate normal variables with correlation ρ = 0.5. Using the cokurtosis definition K(X,X,Y,Y) = E[X^2 Y^2] for zero-mean, unit-variance variables, what is its value?

For standardized bivariate normal variables, E[X²Y²] equals 1 plus 2 times the squared correlation. With correlation 0.5 this is 1 + 2(0.25) = 1.50. Ignoring correlation would give 1.00, which holds only under independence.

  1. A1.50Correct
  2. B1.25
  3. C2.00
  4. D1.00

Explanation

For jointly normal zero-mean, unit-variance variables, E[X^2Y^2] = 1 + 2ρ^2. With ρ = 0.5, ρ^2 = 0.25, so the value is 1 + 0.5 = 1.50. Using 1 + ρ^2 gives 1.25, using 1 + 2ρ gives 2.00, and ignoring correlation (independence) gives 1.00.

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