Verified solution · Exam P
This Exam P sample reference tests Expectation of a Function. This problem can be evaluated without copying the three candidate integrals. Symmetry gives the signed deviation, and a short negative-part correction gives an absolute expectation of 13/12; the matching expression is choice E.
Independent solution
Setup
Rewrite absolute deviation as signed deviation plus twice its negative part.
Model
The density is symmetric around 3, so its mean is 3. Only values below 2 contribute to the correction.
Compute
Evaluate the correction from its polynomial antiderivative, then add it twice to the signed deviation.
Answer
The expression with these three intervals and signs is choice E.
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Original practice · fully worked
A sensor adjustment R takes values -2, 0, 3, and 7 with probabilities 0.15, 0.25, 0.40, and 0.20, respectively. A quality penalty is defined as C=|R-2|. Calculate E[C].
The four penalties are 4, 2, 1, and 5. Weighting them by their probabilities gives 0.15(4)+0.25(2)+0.40(1)+0.20(5)=2.50, so choice C.
Setup
Evaluate the absolute penalty at each possible adjustment value.
Model
Use the expectation-of-a-function formula for a discrete distribution.
Compute
Multiply each penalty by its probability and add the four contributions.
Answer
The expected penalty is 2.50.
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ActuaryProof. “SOA Exam P Sample Question #443 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
