Verified solution · Exam P

SOA Exam P Sample Question #577 Solution

This Exam P sample reference tests Sums of Independent Random Variables. The independent components have combined mean 250 and combined standard deviation √(40²+30²)=50. Dividing 50 by 250 gives coefficient of variation 0.20, so choice A is correct.

Independent analysis by ActuaryProof · reviewed August 4, 2026

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Independent solution

How to solve this Sums of Independent Random Variables question

Setup

Add the two component means to obtain the expected combined loss.

E[S]=100+150=250

Model

Independence makes the component variances additive.

\operatorname{Var}(S)=40^2+30^2=2500

\operatorname{SD}(S)=50

Compute

Form the dimensionless ratio of standard deviation to mean.

\operatorname{CV}(S)=\frac{\operatorname{SD}(S)}{E[S]}

\operatorname{CV}(S)=\frac{50}{250}=0.20

Answer

The combined coefficient of variation is 0.20.

\boxed{0.20\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 adds the component standard deviations to get 70 and divides by 250, instead of adding variances.
CThis averages the two individual coefficients of variation, 0.40 and 0.20, without weighting the independent aggregate correctly.
DThis reports the first component's coefficient of variation, 40/100, rather than that of the sum.
EThis adds the two component coefficients of variation, even though coefficients of variation are not additive.

Original practice · fully worked

Original variant: unknown component spread

Two independent laboratory costs A and B have means 40 and 60. The standard deviation of A is 6, while the coefficient of variation of A+B is 0.10. Calculate the standard deviation of B.

A
4
B
6
C
8
D
10
E
16

Variant answer in brief

The combined mean is 100, so CV 0.10 implies combined standard deviation 10 and variance 100. Independence gives Var(B)=100-36=64, hence SD(B)=8 and choice C is correct.

Setup

Use the combined mean and coefficient of variation to recover the total standard deviation.

E[A+B]=40+60=100

\operatorname{SD}(A+B)=0.10(100)=10

Model

Independent component variances add to the total variance.

10^2=6^2+\operatorname{Var}(B)

Compute

Isolate the unknown variance and take its square root.

\operatorname{Var}(B)=100-36=64

\operatorname{SD}(B)=8

Answer

The unknown component standard deviation is 8.

\boxed{8\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #577 — Verified Solution.” Verified 2026-07-30.

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

How to solve this Sums of Independent Random Variables question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: unknown component spread

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution