Verified solution · Exam P
This Exam P sample reference tests Distribution Functions. This is a distribution-function normalization followed by an upper-tail calculation. The endpoint condition gives c=1/30, the loss distribution has F(3.2)=0.448, and payment occurs in the complementary event with probability 0.552, so choice D is correct.
Independent solution
Setup
A cumulative distribution must reach one at the upper endpoint of its finite support. Use that requirement to determine the constant.
Model
A positive payment under an ordinary deductible occurs exactly when the loss exceeds the deductible.
Compute
Evaluate the distribution function at the deductible and take its complement.
Answer
The insurer makes a positive payment with probability 0.552.
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Original practice · fully worked
A normalized polymer response X has density f(x)=k x(2-x) for 0<x<2 and zero elsewhere. A monitoring system reports Y=(X-1)² rather than X itself. Calculate the probability that the reported value is at most 1/4.
Normalizing the response density gives k=3/4. The event Y at most 1/4 is the central interval 1/2 at most X at most 3/2, whose integrated probability is 11/16=0.6875, so choice D is correct.
Setup
Determine the density constant from its total integral over the support.
Model
Translate the nonmonotone transformation event back into an interval for X.
Compute
Integrate the normalized density across the central interval.
Answer
The transformed response is at most one-quarter with probability 0.6875.
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ActuaryProof. “SOA Exam P Sample Question #609 — Verified Solution.” Verified 2026-07-30.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.
