Worked solution · Exam P

SOA Exam P Sample Question #232 Solution

This Exam P sample reference tests Moments. Expanding the shifted second moment gives a mean of 7.5. The variance is then 61-7.5 squared=4.75, so the standard deviation is 2.18, choice A.

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 Moments question

Setup

Expand the centered expression so that the unknown first moment appears.

\operatorname{E}[(X-1)^2]=\operatorname{E}[X^2]-2\operatorname{E}[X]+1

Model

Insert the two supplied second-moment values and solve for the mean.

47=61-2\operatorname{E}[X]+1

\operatorname{E}[X]=\frac{61+1-47}{2}=7.5

Compute

Convert the raw second moment to variance and take its positive square root.

\operatorname{Var}(X)=61-7.5^2=4.75

\operatorname{SD}(X)=\sqrt{4.75}=2.1794495

Answer

The standard deviation rounds to 2.18.

\boxed{2.18\quad\text{(A)}}

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.

BA standard deviation of 2.40 would imply variance 5.76 and hence E[X squared]=5.76+7.5 squared=62.01, contradicting the supplied second moment 61.
CThe value 7.31 is √(61-7.5). This subtracts the mean rather than the squared mean when converting a raw second moment to variance.
DThe value 7.50 is E[X], recovered from the shifted-moment identity. The question asks for the standard deviation.
EThe value 7.81 is √(61), the root-mean-square value. It omits subtraction of E[X] squared.

Original practice · fully worked

Original variant: source spread from a processed second moment

A processed signal is W=3Z-4 for an underlying random variable Z. The processed signal has mean E[W]=8 and second raw moment E[W²]=169. Calculate the standard deviation of Z.

A
3.416
B
4.000
C
4.333
D
10.247
E
13.000

Answer and calculation

The processed variance is 169-8 squared=105. Since Var(W)=9Var(Z), the source standard deviation is √(105)/3=3.416, choice A.

Setup

Convert the processed signal's raw second moment to its variance.

\operatorname{Var}(W)=\operatorname{E}[W^2]-\operatorname{E}[W]^2

\operatorname{Var}(W)=169-8^2=105

Model

Use variance scaling under the affine processing rule.

\operatorname{Var}(W)=\operatorname{Var}(3Z-4)=9\operatorname{Var}(Z)

\operatorname{Var}(Z)=\frac{105}{9}

Compute

Take the positive square root on the source scale.

\operatorname{SD}(Z)=\sqrt{\frac{105}{9}}=\frac{\sqrt{105}}3=3.4156503

Answer

The source standard deviation rounds to 3.416.

\boxed{3.416\quad\text{(A)}}

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Next steps for Exam P Univariate Random Variables

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How to solve this Moments question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: source spread from a processed second moment

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables