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

Monthly loan losses for a lender are i.i.d. with mean USD 4 million and standard deviation USD 3 million. Using the CLT, what is the approximate probability that the average monthly loss over 36 months exceeds USD 4.75 million? (Use N(0,1) cumulative values: Phi(1.50) = 0.9332, Phi(2.00) = 0.9772, Phi(0.75) = 0.7734.)

The standard error of the average is 3 divided by 6, or 0.5 million. The threshold of 4.75 is 0.75 above the mean, giving z of 1.5. The upper-tail probability under the normal approximation is 1 minus 0.9332, which equals about 0.0668.

  1. A0.0668Correct
  2. B0.0228
  3. C0.2266
  4. D0.0332

Explanation

Standard error = 3 / sqrt(36) = 0.5. Z = (4.75 - 4) / 0.5 = 1.50. P(Z > 1.50) = 1 - 0.9332 = 0.0668. The 0.0228 option wrongly uses z = 2.00 (standard error of 0.375 error), and 0.2266 uses z = 0.75 with the unscaled standard deviation of... 3 /... i.e., ignores sqrt(n) partly.

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