Verified solution · Exam P

SOA Exam P Sample Question #441 Solution

This Exam P sample reference tests Uniform Distribution. The first uniform standard deviation gives b=9.60√(12). The second interval then has width 2b-6, so its standard deviation is (2b-6)/√(12)=17.468, which rounds to choice C.

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.

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Independent solution

How to solve this Uniform Distribution question

Setup

For a continuous uniform variable, standard deviation equals its support width divided by the square root of twelve.

\operatorname{SD}(U)=\frac{w}{\sqrt{12}}

Model

Use the first interval width to recover the unknown endpoint.

\frac{b}{\sqrt{12}}=9.60

b=9.60\sqrt{12}=33.25537551\ldots

Compute

Form the width of the second support and apply the same uniform formula.

w_2=2b-6

\operatorname{SD}_2=\frac{2b-6}{\sqrt{12}}

\operatorname{SD}_2=2(9.60)-\frac6{\sqrt{12}}=19.20-\sqrt3=17.46794919\ldots

Answer

The second standard deviation rounds to 17.5.

\boxed{17.5\quad\text{(C)}}

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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.

AThis applies the endpoint transformation directly to the first standard deviation, calculating 2(9.60) − 6 = 13.2. The subtraction changes the support width before division by √12.
BThis divides the six-unit width adjustment by √3 instead of √12, giving 19.2 − 6 / √3 = 15.736, near 15.7.
DThis ignores the six-unit reduction in the second support width and reports 2(9.60) = 19.2.
EA standard deviation of 19.9 would require a support width 19.9√12 = 68.94, but the recovered width is 2(33.2554) − 6 = 60.51.

Original practice · fully worked

Original variant: transform a symmetric dial error

A dial error U is uniformly distributed on the symmetric interval from -a to a. Its mean absolute error is 4 units. A corrected output is V=3U+2. Calculate the variance of V.

A
8
B
21.33
C
64
D
192
E
576

Variant answer in brief

For a symmetric uniform error, E|U|=a/2, so a=8 and Var(U)=a²⁄³=64/3. Scaling by 3 multiplies variance by 9, giving 192 and choice D.

Setup

Use symmetry to evaluate the mean absolute error in terms of the unknown half-width.

\operatorname{E}[|U|]=2\int_0^a u\frac{1}{2a}\,du=\frac a2

Model

Recover the half-width and then the variance of a uniform variable on the symmetric support.

\frac a2=4\quad\Longrightarrow\quad a=8

\operatorname{Var}(U)=\frac{(2a)^2}{12}=\frac{a^2}{3}=\frac{64}{3}

Compute

Apply the affine variance rule; the additive constant does not affect spread.

\operatorname{Var}(V)=\operatorname{Var}(3U+2)=3^2\operatorname{Var}(U)

\operatorname{Var}(V)=9\left(\frac{64}{3}\right)=192

Answer

The corrected output has variance 192.

\boxed{192\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #441 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

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: transform a symmetric dial error

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution