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

A risk analyst models daily returns as a mixture of two normal distributions that both have a mean of zero but different standard deviations. Which statement about the mixture is correct?

The mixture is symmetric with zero skewness, but its excess kurtosis is positive. Combining normals with the same mean and different variances creates a peaked center and heavier tails than any single normal, so the mixture is not normal despite normal components.

  1. AIt has zero skewness and excess kurtosis greater than zeroCorrect
  2. BIt is itself a normal distribution because both components are normal
  3. CIt has negative skewness because one component has a larger variance
  4. DIt has excess kurtosis of zero because both components have zero excess kurtosis

Explanation

With equal means, the mixture is symmetric, so skewness is zero. Mixing different variances puts more weight in the center and in the tails than a single normal with the same variance, so kurtosis exceeds 3. A mixture of normals with different variances is not normal, and the kurtosis of a mixture is not the weighted average of the component kurtosis values.

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