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

X and Y are independent random variables with zero means and finite fourth moments. What is the standardized cokurtosis K(X,X,Y,Y) = E[X^2 Y^2] / (sigma_X^2 sigma_Y^2)?

For independent variables, the expectation of X squared times Y squared factors into the product of their variances, which exactly cancels the denominator. The standardized cokurtosis K(X,X,Y,Y) therefore equals 1, whatever the individual kurtoses of X and Y are.

  1. A0
  2. B1Correct
  3. C3
  4. DIt depends on the kurtosis of X and of Y

Explanation

Independence gives E[X^2 Y^2] = E[X^2] E[Y^2] = sigma_X^2 sigma_Y^2, so the ratio equals 1 regardless of the marginal distributions. The value 3 is the kurtosis of a normal variable, K(X,X,X,X), not this cross moment. The value 0 would require E[X^2 Y^2] = 0, which is not true for non-degenerate variables.

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