Worked solution · Exam P

SOA Exam P Sample Question #707 Solution

This Exam P sample reference tests Uniform Distribution. Write the claim payment as the positive part of the loss after the deductible. Integrating its triangular tail gives an expected payment of (500-d)²⁄¹⁰⁰⁰; setting this equal to 90 yields d = 200, so the verified answer is choice B.

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 Uniform Distribution question

Setup

Let X be the loss and d the fixed deductible. The insurer pays only the portion of X that exceeds d.

X\sim\operatorname{Uniform}(0,500),\qquad Y=(X-d)_+

Model

Average the positive excess over the uniform density. A deductible in the support leaves a triangular payment region.

\mathbb{E}[Y]=\int_d^{500}(x-d)\frac{1}{500}\,dx

\mathbb{E}[Y]=\frac{(500-d)^2}{1000}

Compute

Equate the stop-loss expectation to the specified expected payment and take the feasible square root.

\frac{(500-d)^2}{1000}=90

(500-d)^2=90{,}000

500-d=300,\qquad d=200

Answer

The required fixed deductible is 200 monetary units.

\boxed{d=200\quad\text{(B)}}

Test the method on unseen problems

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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.

ASubtracting the expected payment directly from the mean loss gives 250 − 90 = 160. That treats every loss as if it exceeded the deductible and ignores zero-payment outcomes.
CAt a deductible of 240, the stop-loss expectation is (500-240)²⁄¹⁰⁰⁰ = 67.6, not 90; choosing a value near the original mean without integrating the tail understates payment truncation.
DOmitting the one-half from the triangular tail area gives (500-d)²⁄⁵⁰⁰ = 90 and d about 287.9, which points toward D but doubles the true expected payment formula.
ESetting the conditional mean excess (500-d)/2 equal to 90 gives d = 320. This omits the 0.36 probability that a loss actually exceeds that deductible.

Original practice · fully worked

Original variant: storm-repair reserve

A municipal arts venue models its next storm-repair cost X as uniform between 200 and 1400 dollars. A reserve reimburses only the portion of a repair cost above a 500-dollar trigger. Calculate the expected reimbursement from the reserve.

A
$225.00
B
$300.00
C
$337.50
D
$450.00
E
$900.00

Answer and calculation

A repair exceeds the trigger with probability 0.75, and its mean excess conditional on doing so is 450 dollars. Their product is 337.50 dollars, making choice C correct.

Setup

Represent the reserve reimbursement as the positive excess of the repair cost over the trigger.

X\sim\operatorname{Uniform}(200,1400),\qquad R=(X-500)_+

Model

A cost above 500 is uniform over the remaining interval, so separate the chance of reimbursement from its conditional amount.

\Pr(X>500)=\frac{1400-500}{1400-200}=\frac{3}{4}

\mathbb{E}[X-500\mid X>500]=\frac{0+900}{2}=450

Compute

Multiply the probability of crossing the trigger by the average excess once it is crossed.

\mathbb{E}[R]=\frac{3}{4}(450)=337.50

Answer

The reserve's expected reimbursement is 337.50 dollars.

\boxed{\mathbb{E}[R]=\$337.50\quad\text{(C)}}

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Next steps for Exam P Univariate Random Variables

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How to solve this Uniform Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: storm-repair reserve

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables