Worked solution · Exam P
This Exam P sample reference tests Density Normalization. This problem normalizes a linearly increasing density and then inverts its CDF. The resulting 80th percentile is √(3.2)=1.78885, which selects choice E.
Independent solution
Setup
Write the density as cx on its support and determine c from total probability.
Model
Integrate the normalized density to obtain the cumulative probability.
Compute
Set the CDF equal to 0.80 and take the positive root within the support.
Answer
The percentile rounds to 1.79.
Test the method on unseen problems
30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
Error diagnosis
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
A laboratory response score X has density c for 0<x<2, density 2c for 2≤x<5, and density zero elsewhere, where c is a positive constant. Calculate the 60th percentile of X.
Normalization gives c=1/8, and the lower segment contains probability 0.25. The remaining 0.35 needed to reach 0.60 lies in the upper segment, giving q=3.4 and choice C.
Setup
Normalize the two constant-density segments.
Model
Determine which segment contains the requested percentile.
Compute
Add the lower-segment mass to the probability accumulated in the upper segment.
Answer
The 60th percentile is 3.4.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 387 Exam P univariate random variables practice problems.
Keep studying
Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.
