Worked solution · Exam P

SOA Exam P Sample Question #284 Solution

This Exam P sample reference tests Exponential Distribution. The exponential expenditure has variance 2000. Correlation 0.20 converts to covariance 1000, so the total variance is 2000+12500+2(1000)=16500 and choice D.

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 Exponential Distribution question

Setup

Read the exponential scale as its mean and square it to obtain the first expenditure variance.

E[X]=20\sqrt5

\operatorname{Var}(X)=(20\sqrt5)^2=2000

Model

Convert the supplied correlation to covariance using the two standard deviations.

\operatorname{Cov}(X,Y)=0.20\sqrt{2000}\sqrt{12500}

=0.20(5000)=1000

Compute

Expand the variance of the total, including both covariance cross terms.

\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)

=2000+12500+2(1000)=16500

Answer

The total-expenditure variance is 16,500.

\boxed{16500\quad\text{(D)}}

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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 12,500 reports Var(Y) alone and omits both the exponential expenditure variance and the positive covariance contribution.
BThe value 13,500 is 12,500+1,000. It adds one covariance but omits Var(X) and the second covariance cross term.
CThe value 15,500 is 2,000+12,500+1,000. It includes both marginal variances but adds Cov(X,Y) only once instead of twice.
EUsing the square root of the correlation in the covariance formula gives Cov=sqrt[0.20(2000)(12500)]=2236.068. Then 2000+12500+2(2236.068)=18972.136, producing this choice.

Original practice · fully worked

Original variant: correlation between a balance and a contrast

Daily warehouse demand has two centered shocks. Shock X has variance 4 square units and shock Y has variance 25 square units, with correlation 0.30 between them. The warehouse tracks the balance U=X+Y and the contrast V=X-Y. Calculate Corr(U,V).

A
-21.000
B
-0.913
C
-0.740
D
0.300
E
0.740

Answer and calculation

The original covariance is 3. The balance and contrast have variances 35 and 23, while their covariance is Var(X)-Var(Y)=-21. Normalization gives -21/√(805)=-0.740153 and choice C.

Setup

Convert the original correlation into covariance.

\operatorname{Cov}(X,Y)=0.30(2)(5)=3

Model

Expand the two transformed variances.

\operatorname{Var}(U)=4+25+2(3)=35

\operatorname{Var}(V)=4+25-2(3)=23

Compute

Find the covariance of the balance and contrast, then normalize it.

\operatorname{Cov}(U,V)=\operatorname{Cov}(X+Y,X-Y)=4-25=-21

\operatorname{Corr}(U,V)=\frac{-21}{\sqrt{35}\sqrt{23}}=-0.7401527457\ldots

Answer

The balance and contrast have correlation approximately -0.740.

\boxed{-0.740\quad\text{(C)}}

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How to solve this Exponential Distribution question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: correlation between a balance and a contrast

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables