Worked solution · Exam P
This Exam P sample reference tests Continuous Distributions. Normalizing the power density gives cumulative distribution F(x)=x⁶. The probability that neither of two independent losses exceeds the threshold is therefore d¹². Setting its complement equal to P gives d=(1-P)⁽¹⁄¹²⁾, so choice E is correct.
Independent solution
Setup
Normalize the power density and integrate it to obtain the loss distribution function.
Model
No payment on either independent observation means both values are at most the threshold.
Compute
Complement the no-payment event and solve for the threshold.
Answer
The required expression is the twelfth root of one minus P.
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 coating score X has density f(x)=2(1-x) for 0<x<1. Four scores are sampled independently, and M is their minimum. Calculate the 25th percentile of M.
One score has survival function (1-x)², so the four-score minimum has survival function (1-m)⁸. Setting its CDF to 0.25 gives m=1-0.75⁽¹⁄⁸⁾=0.035321, so choice A is correct.
Setup
Integrate the density and write the survival function of one score.
Model
The minimum exceeds m only if all four independent scores exceed m.
Compute
Set the minimum's CDF equal to one quarter and solve.
Answer
The lower quartile of the minimum rounds to 0.0353.
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