Worked solution · Exam P

SOA Exam P Sample Question #586 Solution

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 analysis by ActuaryProof · reviewed July 30, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Exponential Distribution question

Setup

Work in thousands and express the insurer payment as a positive-part transformation of the loss L.

Y=0.8(L-0.5)_+

\Pr(Y>5)=\Pr\!\left(0.8(L-0.5)>5\right)

Model

Solve the payment inequality for the underlying loss threshold.

L-0.5>\frac{5}{0.8}=6.25

L>6.75

Compute

Use the exponential survival function with scale 6 at the transformed threshold.

\Pr(L>\ell)=e^{-\ell/6}

\Pr(Y>5)=e^{-6.75/6}=e^{-1.125}=0.3246524674\ldots

Answer

The payment exceeds the specified amount with probability approximately 0.325.

\boxed{0.325\quad\text{(C)}}

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Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis reflects a severe unit or threshold conversion error; the payment condition should produce a loss threshold of 6.75 in the model's thousand-unit scale.
BThis models the payment as 0.8L-0.5, applying coinsurance before subtracting the deductible, rather than 0.8(L-0.5).
DThis is the complement of the correct exponential tail and therefore represents the probability that the payment does not exceed the threshold.
EThis is the complement of the value generated by the incorrect transformation 0.8L-0.5, combining both a payment-model error and a reversed tail.

Original practice · fully worked

Original variant: conditional repair-payment percentile

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.

A
2.400
B
4.926
C
5.526
D
6.126
E
9.210

Answer and calculation

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.

R=0.6(X-1)_+

R>0\quad\Longleftrightarrow\quad X>1

Model

Memorylessness makes the deductible excess exponential with the original mean; multiplying by 0.6 rescales that mean.

X-1\mid(X>1)\sim\operatorname{Exponential}(\text{mean }4)

R\mid(R>0)\sim\operatorname{Exponential}(\text{mean }2.4)

Compute

At the 90th percentile q, the conditional survival probability is 0.10.

e^{-q/2.4}=0.10

q=2.4\ln(10)=5.526204223\ldots

Answer

The conditional 90th percentile of the payment is approximately 5.526 thousand dollars.

\boxed{5.526\text{ thousand dollars}\quad\text{(C)}}

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How to solve this Exponential Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: conditional repair-payment percentile

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables