Worked solution · Exam P
This Exam P sample reference tests Loss Models. The retained loss is min(X,2). Integrating x below the deductible and 2 above it gives mean 0.93427, choice C.
Independent solution
Setup
The retained loss is the smaller of the original loss and the deductible level two.
Model
Compute its expectation by integrating the loss amount below two and adding the pointwise contribution of two for losses above two.
Compute
Evaluating the integral and tail probability gives mean retained loss approximately 0.934273.
Answer
The mean retained loss is approximately 0.93, selecting choice C.
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.
Original practice · fully worked
A load X has density f(x)=3/x⁴ for x at least 1. A relief contract pays the amount by which X exceeds 2. Find the expected payment.
The Pareto survival function is x⁻³. Integrating that tail beyond two gives expected excess 1/8=0.125, choice E.
Setup
Integrating the Pareto-type density gives a survival function equal to the inverse cube of the threshold.
Model
The expected excess above two is the integral of that survival function from two to infinity.
Compute
The tail integral equals one eighth, or 0.125.
Answer
The expected excess payment is 0.125, selecting choice E.
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