Verified solution · Exam P
This Exam P sample reference tests Density Normalization. Normalizing the density gives a survival function of (1+x²)⁽⁻⁸⁾. Setting that survival probability to 0.98 gives the 2nd percentile 0.050284, so choice D is correct.
Independent solution
Setup
Write the nonnegative density with an unknown normalizing constant and normalize it by substituting u=1+x².
Model
Integrating the normalized density gives a convenient closed form for the distribution and survival functions.
Compute
At the 2nd percentile, the survival probability is 0.98. Invert the survival function.
Answer
The percentile rounds to 0.050.
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Original practice · fully worked
A component's nonnegative stress index Y has survival function S(y)=(1+y)⁽⁻³⁾. Inspection shows that the component has already exceeded stress level 1. Calculate the conditional median of the remaining excess Y-1.
Conditioning above level 1 changes the excess survival function to [2/(2+m)]³. Setting it equal to one half gives m=2(2⁽¹⁄³⁾-1)=0.519842, so choice C is correct.
Setup
Let R=Y-1 after conditioning on Y>1. Express the tail of R as a ratio of the original survival probabilities.
Model
A continuous conditional median leaves one half of the conditional distribution above it.
Compute
Invert the conditional survival equation.
Answer
The conditional median remaining excess rounds to 0.520.
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ActuaryProof. “SOA Exam P Sample Question #508 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
