Verified solution · Exam P
This Exam P sample reference tests Exponential Distribution. In thousands, a payment above 5 requires the loss excess over the 0.5 deductible to exceed 5/0.8=6.25. The loss must therefore exceed 6.75, whose exponential survival probability is exp(-6.75/6)=0.32465, so choice C is correct.
Independent solution
Setup
Work in thousands and express the insurer payment as a positive-part transformation of the loss L.
Model
Solve the payment inequality for the underlying loss threshold.
Compute
Use the exponential survival function with scale 6 at the transformed threshold.
Answer
The payment exceeds the specified amount with probability approximately 0.325.
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Original practice · fully worked
A repair cost X follows an exponential distribution with mean 4 thousand dollars. A service plan pays 60% of the amount above a 1-thousand-dollar deductible. Given that the plan makes a positive payment, calculate the 90th percentile of the payment, in thousands.
Given a positive payment, exponential memorylessness makes the excess over the deductible exponential with mean 4. Multiplying by 0.60 gives a conditional payment with mean 2.4. Its 90th percentile is 2.4ln(10)=5.526, choice C.
Setup
Let R be the service-plan payment and condition on the event that the repair cost clears the deductible.
Model
Memorylessness makes the deductible excess exponential with the original mean; multiplying by 0.6 rescales that mean.
Compute
At the 90th percentile q, the conditional survival probability is 0.10.
Answer
The conditional 90th percentile of the payment is approximately 5.526 thousand dollars.
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ActuaryProof. “SOA Exam P Sample Question #586 — Verified Solution.” Verified 2026-07-30.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.
