Verified solution · Exam P

SOA Exam P Sample Question #668 Solution

This Exam P sample reference tests Uniform Distribution. For a uniform interval [a,b], the median condition gives a+b=12 and the 90th-percentile condition gives a+0.9(b-a)=13.2. Thus a=-3 and b=15, yielding second moment 63 and choice E.

Independent analysis by ActuaryProof · reviewed August 4, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

Read the solution Open official source

Independent solution

How to solve this Uniform Distribution question

Setup

Let a and b be the lower and upper endpoints. Uniform percentiles advance linearly across the interval.

q_p=a+p(b-a)

Model

Translate the median and 90th-percentile information into two endpoint equations.

\frac{a+b}{2}=6

a+0.9(b-a)=13.2

Compute

Solve the two equations, then use the raw second-moment formula for a continuous uniform distribution.

a=-3,\qquad b=15

\operatorname{E}[X^2]=\frac{a^2+ab+b^2}{3}

\operatorname{E}[X^2]=\frac{9-45+225}{3}=63

Answer

The second raw moment equals 63.

\boxed{63\quad\text{(E)}}

Continue without hunting through PDFs

All 718 Exam P sample solutions in syllabus order

The searchable 3108-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam P study manual PDF

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis reports the median, which is also the uniform mean, rather than the second raw moment.
BThis is the variance (15-(-3)) squared divided by 12, so it omits the squared mean needed for the second moment.
CThis adds the mean 6 to the variance 27; the second-moment identity requires the squared mean 36 instead.
DThis is the squared mean 6 squared and omits the variance 27.

Original practice · fully worked

Original variant: calibration-error interval

A calibration error U is modeled as continuous uniform on an unknown interval. Its 25th percentile is -2 units and its 75th percentile is 6 units. Calculate the second raw moment of U.

A
2.000
B
4.000
C
16.000
D
21.333
E
25.333

Variant answer in brief

The two quartile equations imply endpoints -6 and 10. The variance is 64/3 and the squared mean is 4, so the second raw moment is 76/3=25.333 and choice E.

Setup

Let c and d denote the unknown endpoints and write the linear uniform quantile formula.

q_p=c+p(d-c)

Model

Apply the formula to the two stated quartiles.

c+0.25(d-c)=-2

c+0.75(d-c)=6

Compute

Solve for the endpoints and substitute them into the uniform second-moment formula.

c=-6,\qquad d=10

\operatorname{E}[U^2]=\frac{c^2+cd+d^2}{3}

\operatorname{E}[U^2]=\frac{36-60+100}{3}=\frac{76}{3}

Answer

The second raw moment is approximately 25.333 square units.

\boxed{25.333\quad\text{(E)}}

Keep practicing

Turn this solution into a complete study sequence

The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

See the Exam P study manual PDF — $49 More Exam P Univariate Random Variables practice problems Free Exam P formula sheet

Cite this solution

ActuaryProof. “SOA Exam P Sample Question #668 — Verified Solution.” Verified 2026-07-30.

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

How to solve this Uniform Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: calibration-error interval

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution