Worked solution · Exam P
This Exam P sample reference tests Continuous Distributions. Normalizing the power density gives a CDF proportional to x raised to a+1. The supplied CDF value calibrates that exponent near 2.5, and the upper tail beyond 4 is 0.4276, choice B.
Independent solution
Setup
Normalize the density over its finite support before using the probability statement.
Model
After normalization, the CDF depends only on the ratio of the point to the upper endpoint.
Compute
Recover the power from logarithms and evaluate the complementary CDF at 4.
Answer
The requested upper-tail probability rounds to 0.428.
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 quality score X lies between 0 and 12 and has cumulative distribution F(x)=(x/12)ᵏ on that interval, where k is positive. Among scores no greater than 9, the fraction no greater than 6 is 4/9. Determine the median score.
The conditional fraction gives (2/3)ᵏ=4/9, so k=2. Solving F(m)=1/2 then gives m=12/√(2)=8.485, choice C.
Setup
Express the supplied conditional fraction as a ratio of CDF values.
Model
Substitute the power-form CDF; the common endpoint cancels.
Compute
The calibration gives k=2, after which the median equation is immediate.
Answer
The median quality score is approximately 8.485.
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