Worked solution · Exam P
This Exam P sample reference tests Continuous Random Variables. Integrate the density shape and divide by its total integral. Setting the resulting CDF equal to one-half gives m=√(√(26)-1)=2.0246, which matches choice A.
Independent solution
Setup
The unknown proportionality constant cancels when the partial integral is divided by the full-support integral.
Model
Form the CDF on the support from the ratio of accumulated density shape to total density shape.
Compute
A median divides the probability mass into equal halves.
Answer
The closest listed value is 2.02.
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 laboratory sensor reports a response level Z between 0 and 3. Its probability density is proportional to 1+z on that interval and is zero elsewhere. Calculate the 80th percentile of Z.
The accumulated density shape is z+z²⁄² and its total is 7.5. Setting the ratio equal to 0.80 gives z=-1+√(13)=2.6056, so choice D is correct.
Setup
Integrate the density shape over the entire support.
Model
Express the CDF as a ratio so the proportionality constant cancels.
Compute
Set the CDF equal to 0.80 and solve the admissible quadratic root.
Answer
The result rounds to the value under choice D.
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