Worked solution · Exam P
This Exam P sample reference tests Insurance Payment Variables. The density corresponds to a Pareto tail S(x)=(1000/x)³ above 1000. The expected positive excess over 1100 is the integral of this survival function, equal to 1000³ divided by 2(1100)², or 413.2231, so choice C is correct.
Independent solution
Setup
Integrate the density once to express the loss survival function above its lower endpoint.
Model
For an ordinary deductible, the expected positive excess equals the tail integral beginning at the deductible.
Compute
Evaluate the power-tail integral.
Answer
The expected payout rounds to the listed value 413.
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 battery endurance X has survival function 1 for 0≤x<200 and (200/x)² for x≥200. A test rig records the smaller of X and an upper display limit L. If the expected displayed reading is 360, determine L.
The limited expectation is the survival integral through L, equal to 400-40000/L. Setting this expression to 360 gives L=1000, so choice D is correct.
Setup
Express a capped nonnegative expectation as an integral of the survival function up to the cap.
Model
Split the integral at the Pareto lower endpoint.
Compute
Equate the limited expectation to the observed display mean and solve for the upper limit.
Answer
The display limit is 1000 endurance units.
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