Verified solution · Exam P
This Exam P sample reference tests Continuous Random Variables. Normalizing the inverse-square kernel over its support gives constant 25/2. Integrating the resulting density from zero to five gives 5/12=0.416667, so the listed answer is choice C.
Independent solution
Setup
Write the density as an unknown constant times the supplied kernel on its finite support.
Model
Normalize the density to determine the proportionality constant.
Compute
Integrate the normalized density over the requested lower interval.
Answer
The probability rounds to 0.42.
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Original practice · fully worked
A quality index Q has density f(q)=c(1+q) for 0<q<2 and zero elsewhere. Calculate the median of Q.
Normalization gives c=1/4. Setting the cumulative probability at m equal to one-half yields m+m squared/2=2, whose positive solution is √(5)-1=1.236068 and choice C.
Setup
Normalize the rising linear density on its support.
Model
A continuous median m divides the normalized probability into equal halves.
Compute
Solve the resulting quadratic and retain the root inside the support.
Answer
The median rounds to 1.236.
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ActuaryProof. “SOA Exam P Sample Question #297 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
