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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

Let Z1, Z2, Z3 and Z4 be independent standard normal random variables. Define W = Z1^2 + Z2^2 + Z3^2 + Z4^2. What is the mean and variance of W?

W follows a chi-squared distribution with 4 degrees of freedom, so its mean equals 4 and its variance equals twice the degrees of freedom, which is 8. The sum of squared independent standard normals defines the chi-squared distribution.

  1. AMean 4, variance 8Correct
  2. BMean 4, variance 4
  3. CMean 8, variance 4
  4. DMean 2, variance 8

Explanation

W is chi-squared with k = 4 degrees of freedom. Mean = k = 4 and variance = 2k = 8. Variance 4 confuses variance with the mean; the others misstate k.

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