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FRM Part I · FRM Exam Part I · Fundamentals of Probability

A continuous random variable X has CDF F(x) = 0 for x < 0, F(x) = x^2/4 for 0 ≤ x ≤ 2, and F(x) = 1 for x > 2. A risk analyst wants the 75th percentile of X and the density at that point. Which pair is correct?

Set the CDF x squared over 4 equal to 0.75, giving x = √3, about 1.732. The density is the derivative of the CDF, x/2, which at √3 equals about 0.866. The 1.5 answer wrongly assumes a uniform distribution.

  1. APercentile = 1.5; density = 0.75
  2. BPercentile = √3 ≈ 1.732; density = 0.866Correct
  3. CPercentile = √3 ≈ 1.732; density = 0.75
  4. DPercentile = 1.5; density = 0.866

Explanation

Solve x^2/4 = 0.75, so x^2 = 3 and x = √3 ≈ 1.732. The density is f(x) = F'(x) = x/2 = 0.866 at that point. Choosing 1.5 comes from treating 0.75 × 2 as the percentile, as if X were uniform; 0.75 is the probability, not the density.

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