SOA Exam P Sample Question #707. Official problem wording is withheld; follow the source link to read it.

SOA Exam P Sample Question #707 — Verified Solution

Independent solution

How to solve this Uniform Distribution question

Answer in brief

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.

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)}}

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

Variant answer in brief

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)}}

Continue by topic

Build a complete proof notebook

The Probability Proof Manual reorganizes verified solutions by syllabus skill and adds formula proofs, error patterns, and original practice.

Explore study products Get the free formula sheet

Cite this solution

ActuaryProof. “SOA Exam P Sample Question #707 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.