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FRM Part I · FRM Exam Part I · Machine-Learning Methods

A bagged ensemble averages B identical-variance tree predictions. Each tree has prediction variance 0.40, and the pairwise correlation between any two trees' predictions is 0.25. Using the variance of an average of B equally correlated predictions, what is the ensemble variance as B becomes very large, and by what approximate fraction does it reduce the variance of a single tree?

The limiting ensemble variance is 0.10, a 75% reduction from 0.40. Averaging removes the independent part of the variance, but the correlated part, rho times sigma squared, equals 0.25 times 0.40, and remains no matter how many trees are added.

  1. A0.00; a 100% reduction
  2. B0.10; a 75% reductionCorrect
  3. C0.10; a 25% reduction
  4. D0.30; a 25% reduction

Explanation

Variance of the average = rho·sigma² + (1-rho)·sigma²/B. As B grows the second term vanishes, leaving 0.25 × 0.40 = 0.10. Reduction from 0.40 is 0.30/0.40 = 75%. Option 0.00 ignores correlation; 0.30 uses (1-rho)·sigma² as the limit, which is the term that vanishes.

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