FRM Part I · FRM Exam Part I · Random Variables
A continuous random variable X has CDF F(x) = 1 − e^(−x/10) for x ≥ 0. What is the 90th percentile of X, closest to?
Set the CDF equal to 0.90, giving e^(−x/10) = 0.10. Then x = 10 × ln(10), which is about 23.0. The quantile is found by inverting the CDF at the stated probability.
- A10.5
- B16.1
- C23.0Correct
- D30.0
Explanation
Solve 1 − e^(−x/10) = 0.90, so e^(−x/10) = 0.10 and x = 10 × ln(10) = 10 × 2.3026 = 23.03. Using ln(1/0.9) gives about 1.05 × 10 = 10.5, which solves for the 10th percentile mistake of using 0.9 inside the exponential. 16.1 comes from the 80th percentile.
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