Worked 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.
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
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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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