Worked solution · Exam P

SOA Exam P Sample Question #252 Solution

This Exam P sample reference tests Sums of Independent Random Variables. Writing the common mean as mu converts the two coefficients of variation into standard deviations 3mu and 4mu. The average retains mean mu and has standard deviation 5mu/2, so its coefficient of variation is 5/2 and choice C.

Independent analysis by ActuaryProof · reviewed July 31, 2026

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Independent solution

How to solve this Sums of Independent Random Variables question

Setup

Translate each coefficient of variation into a standard deviation using the common positive mean.

E[X]=E[Y]=\mu

\operatorname{SD}(X)=3\mu,\qquad \operatorname{SD}(Y)=4\mu

Model

Find the mean and variance of the average, using independence for the variance sum.

E\left[\frac{X+Y}{2}\right]=\mu

\operatorname{Var}\left(\frac{X+Y}{2}\right)=\frac{9\mu^2+16\mu^2}{4}=\frac{25\mu^2}{4}

Compute

Take the square root and divide by the mean.

\operatorname{SD}\left(\frac{X+Y}{2}\right)=\frac{5\mu}{2}

\operatorname{CV}\left(\frac{X+Y}{2}\right)=\frac{5\mu/2}{\mu}=\frac52

Answer

The coefficient of variation is 2.5.

\boxed{\frac52\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.

AThe value 5/4 takes the root-sum-square value 5 and divides by four. The factor one-half has already scaled both the mean and standard deviation; it must not be applied a second time to the ratio.
BThe value 7/4 adds the two input coefficients of variation and divides by four. Independent variances combine by the root-sum-square rule, not by adding coefficients directly.
DThe value 7/2 averages the two coefficients as (3+4)/2. This ignores variance addition and uses a simple arithmetic average.
EThe value 7 is 3+4. It adds the two coefficients of variation without accounting for either independence or the mean of the combined variable.

Original practice · fully worked

Original variant: coefficient of variation with shared demand

Independent components C, E1, and E2 have respective means 10, 4, and 6 and standard deviations 2, 1, and 3. Two regional loads are A=C+E1 and B=C+E2, so they share the common component C. Calculate the coefficient of variation of A+B.

A
0.105
B
0.141
C
0.170
D
0.267
E
0.340

Answer and calculation

The total is A+B=2C+E1+E2, with mean 30 and variance 26. Its coefficient of variation is √(26)/30=0.169967, which rounds to choice C.

Setup

Rewrite the total in terms of the three independent underlying components.

A+B=2C+E_1+E_2

Model

Apply the repeated common-component coefficient to both the mean and variance.

E[A+B]=2(10)+4+6=30

\operatorname{Var}(A+B)=2^2(2^2)+1^2+3^2=26

Compute

Divide the total standard deviation by its mean.

\operatorname{CV}(A+B)=\frac{\sqrt{26}}{30}=0.1699673171\ldots

Answer

The coefficient of variation rounds to 0.170.

\boxed{0.170\quad\text{(C)}}

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How to solve this Sums of Independent Random Variables question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: coefficient of variation with shared demand

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables