Worked solution · Exam P

SOA Exam P Sample Question #181 Solution

This Exam P sample reference tests Uniform Distribution. The reimbursement has a 25% atom at zero. Its 20th percentile is therefore zero, while its median corresponds to the original loss median 500 and equals 500-250=250. The difference is 250, choice B.

Independent analysis by ActuaryProof · reviewed July 31, 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

Write the reimbursement as the positive part of the loss above the deductible.

Y=(X-250)_+,\qquad X\sim\operatorname{Uniform}(0,1000)

Model

Identify the atom at zero and the CDF over positive reimbursement values.

\Pr(Y=0)=\Pr(X\le250)=0.25

F_Y(y)=\Pr(X\le y+250)=\frac{y+250}{1000},\qquad 0<y<750

Compute

Read the lower percentile from the atom and solve the median equation.

q_{0.20}(Y)=0

\frac{q_{0.50}(Y)+250}{1000}=0.50

q_{0.50}(Y)=250

q_{0.50}(Y)-q_{0.20}(Y)=250

Answer

The requested percentile difference is 250.

\boxed{250\quad\text{(B)}}

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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.

AThe value 225 is the difference between the conditional-on-positive median 375 and conditional-on-positive 20th percentile 150. The question includes all losses, including the zero-payment atom.
CThe value 300 is the original loss-quantile difference 500-200. It overlooks that the deductible collapses the entire lowest 25% of losses to a reimbursement of zero.
DThe value 375 is the median reimbursement conditional on a positive reimbursement. Conditioning removes the zero-payment losses and changes the target population.
EThe value 500 is the median loss before applying the deductible. It is not the median reimbursement or the requested difference.

Original practice · fully worked

Original variant: expected reimbursement from a capped layer

A repair cost X is uniformly distributed from 0 to 120 credits. A plan reimburses 50% of the portion of X between 20 and 80 credits, so its maximum reimbursement is 30 credits. Calculate the expected reimbursement.

A
17.500 credits
B
20.833 credits
C
22.500 credits
D
30.000 credits
E
35.000 credits

Answer and calculation

The plan pays half of a 60-credit layer above 20. The layer's expected amount is the survival integral from 20 to 80, equal to 35, so the expected reimbursement is 17.5 credits and choice A.

Setup

Express the contract as coinsurance applied to a limited loss layer.

Y=0.50\min\left((X-20)_+,60\right)

Model

Use a survival integral over the attachment-to-limit interval.

E[Y]=0.50\int_{20}^{80}\Pr(X>t)\,dt

\Pr(X>t)=1-\frac{t}{120}

Compute

Integrate the uniform survival function and apply the coinsurance factor.

E[Y]=0.50\int_{20}^{80}\left(1-\frac{t}{120}\right)dt

=0.50\left[t-\frac{t^2}{240}\right]_{20}^{80}

=0.50(35)=17.5

Answer

The expected reimbursement is 17.5 credits.

\boxed{17.500\quad\text{credits (A)}}

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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: expected reimbursement from a capped layer

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables