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

Which statement best explains why a mixture of two normal distributions with the same mean but different variances exhibits excess kurtosis?

Excess kurtosis arises because the wide component adds extra tail probability while the narrow component concentrates mass near the center, compared with a single normal of the same total variance. The mixture stays symmetric with the same mean, so the effect is purely from fourth-moment heaviness.

  1. AThe mixture has a nonzero skewness that inflates the fourth moment
  2. BThe mixture's mean is shifted away from the component means
  3. CThe high-variance component places more probability in the tails than a single normal with the same overall variance, while the center is also more concentratedCorrect
  4. DThe mixture variance is always smaller than the average of the component variances

Explanation

With equal means, the mixture is symmetric, so skewness is zero and the mean is unchanged. The mixing of a narrow and a wide component gives a more peaked center and heavier tails than a normal with matching variance, which yields kurtosis above 3. The variance of the mixture equals the weighted average, not less.

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