Worked solution · Exam P

SOA Exam P Sample Question #552 Solution

This Exam P sample reference tests Uniform Distribution. A uniform percentile lies the stated fraction of the interval width above the lower endpoint. At probability 0.80 this is a+0.8(b-a)=0.2a+0.8b, which 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

For a uniform variable on the interval from a to b, write the distribution function inside its support.

F(x)=\frac{x-a}{b-a},\qquad a\le x\le b

Model

Let q denote the requested percentile and set its cumulative probability equal to 0.80.

\frac{q-a}{b-a}=0.80

Compute

Move 80% of the interval width upward from the lower endpoint and collect endpoint coefficients.

q=a+0.80(b-a)

q=0.20a+0.80b

Answer

The lower and upper endpoints receive weights 0.20 and 0.80, respectively.

\boxed{0.20a+0.80b\quad\text{(B)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

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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 expression a+0.8b adds 80% of the upper endpoint instead of 80% of the interval width b-a, so it double-counts part of the lower-endpoint shift.
CThe expression 0.8a+0.2b is a+0.2(b-a), the 20th percentile; it reverses the endpoint weights.
DThe expression 0.8(b-a) measures the correct distance from the lower endpoint but omits the required starting value a.
EThe expression 0.8(a+b) scales the sum of the endpoints. It fails the translation check because shifting both endpoints by c would shift it by 1.6c rather than c.

Original practice · fully worked

Original variant: calibrating an unknown uniform range

A machine's calibration score is uniformly distributed on an unknown interval. Its 25th percentile is 18 and its 75th percentile is 34. Calculate the probability that a calibration score exceeds 30.

A
0.125
B
0.250
C
0.375
D
0.625
E
0.750

Answer and calculation

The 16-point interquartile gap is half the full uniform width, so the support is [10,42]. The tail above 30 has length 12 out of 32, giving 0.375 and choice C.

Setup

For a uniform distribution, a percentile difference equals the same fraction of the full support width.

q_{0.75}-q_{0.25}=0.50(U-L)

Model

Recover the width and then the two endpoints from either stated percentile.

34-18=0.50(U-L)\Rightarrow U-L=32

18=L+0.25(32)\Rightarrow L=10,\quad U=42

Compute

Uniform probability is proportional to interval length.

\Pr(X>30)=\frac{42-30}{42-10}=\frac{12}{32}=0.375

Answer

The upper-tail probability is three-eighths.

\boxed{0.375\quad\text{(C)}}

Keep practicing

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: calibrating an unknown uniform range

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables