Worked 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.
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 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.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 387 Exam P univariate random variables practice problems.
Keep studying
Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.
