Worked solution · Exam P
This Exam P sample reference tests Cumulative Distribution Functions. Right continuity requires evaluating the cdf at the threshold with its upper branch, which gives F(V)=0.90. Therefore the strict upper tail is 1-F(V)=0.10, choice B.
Independent solution
Setup
A strict upper-tail probability uses the cdf value at the boundary, not the left-hand limit.
Model
At the boundary, the branch defined for values at least V applies.
Compute
Evaluate the exponential factor and take the complement.
Answer
The strict upper-tail probability is 0.10.
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
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Original practice · fully worked
A service credit X before adjustment is uniformly distributed between 0 and 200 credits. The issued credit is Y=max(X,100), so every value below the guarantee is raised to 100. Calculate Var(Y).
The floor creates probability one half at 100 and leaves the original uniform density above 100. Direct first and second moments are 125 and 50000/3, so Var(Y)=50000/3-125²=3125/3=1041.67, choice B.
Setup
The floor collapses the lower half of the uniform distribution into an atom at 100.
Model
Combine the atom with the unchanged uniform density on the interval above the floor.
Compute
Evaluate the two moments and center the second moment.
Answer
The variance of the issued credit is approximately 1041.67 square credits.
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