Worked solution · Exam P

SOA Exam P Sample Question #504 Solution

This Exam P sample reference tests Uniform Distribution. The retained amount is 40% of the original uniform loss. Its 30th percentile is 0.40 × 4.40, or 1.76, so choice A is correct.

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

Let X be the original loss and U the portion not reimbursed. The reimbursement removes 60%, leaving a positive linear transformation of X.

X\sim\operatorname{Uniform}(2,10)

U=X-0.60X=0.40X

Model

A positive scale factor preserves percentile ranks. First locate the requested quantile of X, then apply the scale factor.

Q_X(p)=2+p(10-2)

Q_U(p)=0.40Q_X(p)

Compute

Evaluate the 30th percentile before and after the transformation.

Q_X(0.30)=2+0.30(8)=4.40

Q_U(0.30)=0.40(4.40)=1.76

Answer

The 30th percentile of the unreimbursed amount is 1.76.

\boxed{1.76\quad\text{(A)}}

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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.

BThis uses 0.30(0.60)(10)=1.80, multiplying the percentile rank, reimbursement rate, and upper endpoint while ignoring both the lower endpoint and the unreimbursed share.
CThis computes the reimbursed rather than retained amount: 0.60Q_X(0.30)=0.60(4.40)=2.64.
DThis reverses the percentile rank and uses the 70th percentile of U: 0.80+0.70(4.00-0.80)=3.04.
EThis stops at the original-loss percentile Q_X(0.30)=2+0.30(8)=4.40 and never applies the 0.40 retained fraction.

Original practice · fully worked

Original variant: reverse calibration quantile

An optical scanner's raw calibration score S is uniform, but its endpoints are not listed. Engineers report that Q_S(0.20)=18 and Q_S(0.80)=42. The display converts the score to D=50-0.5S. Determine the 25th percentile of D.

A
20
B
25
C
30
D
35
E
40

Answer and calculation

The two supplied quantiles recover S as uniform on [10,50]. Because the display transformation is decreasing, its 25th percentile uses the 75th percentile of S and equals 30, so choice C is correct.

Setup

Write S as uniform on [a,a+d]. The gap between the reported percentile ranks determines the support width.

42-18=(0.80-0.20)d

d=40,\qquad a=18-0.20(40)=10

Model

The display is a decreasing affine transformation, so lower display quantiles correspond to upper raw-score quantiles.

Q_D(p)=50-0.5Q_S(1-p)

Compute

Find the raw 75th percentile and transform it.

Q_S(0.75)=10+0.75(40)=40

Q_D(0.25)=50-0.5(40)=30

Answer

The display's 25th percentile is 30.

\boxed{30\quad\text{(C)}}

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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: reverse calibration quantile

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables