Worked solution · Exam P

SOA Exam P Sample Question #253 Solution

This Exam P sample reference tests Variance of Linear Combinations. The additive constant contributes no variance, and independence removes the covariance term. Squaring the two random coefficients gives 9(3)+4(4)=43, so choice D is correct.

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 Variance of Linear Combinations question

Setup

Discard the fixed shift when computing variance.

\operatorname{Var}(3X+2Y-5)=\operatorname{Var}(3X+2Y)

Model

Expand the linear-combination variance and use independence.

\operatorname{Var}(3X+2Y)=3^2\operatorname{Var}(X)+2^2\operatorname{Var}(Y)+12\operatorname{Cov}(X,Y)

\operatorname{Cov}(X,Y)=0

Compute

Insert the two component variances.

\operatorname{Var}(Z)=9(3)+4(4)=27+16=43

Answer

The score variance is 43.

\boxed{43\quad\text{(D)}}

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30 original multivariate 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.

AThe value 12 is Var(X)+Var(Y)+5=3+4+5. It ignores both random coefficients and incorrectly lets the fixed shift contribute.
BThe value 17 is 3Var(X)+2Var(Y)=3(3)+2(4). It scales variances by the coefficients instead of their squares.
CThe value 38 is 43-5. Subtracting a constant shifts the mean but does not subtract that constant from the variance.
EThe value 68 is 43+5 squared. It treats the fixed value -5 as an independent random term with variance 25.

Original practice · fully worked

Original variant: maximum variance under unknown dependence

The marginal variances of two calibration errors are Var(X)=4 and Var(Y)=25. Their correlation is not specified and may take any feasible value from -1 through 1. An index is defined as Z=3X-2Y+7. Calculate the largest possible value of Var(Z).

A
16
B
100
C
120
D
136
E
256

Answer and calculation

Writing Cov(X,Y)=10rho gives Var(Z)=136-120rho. This is largest at rho=-1, where the variance is 256, so choice E.

Setup

Express covariance through the unknown correlation and the two standard deviations.

\operatorname{Cov}(X,Y)=\rho(2)(5)=10\rho

Model

Expand the variance, retaining the signed covariance cross term.

\operatorname{Var}(Z)=3^2(2^2)+(-2)^2(5^2)+2(3)(-2)(10\rho)

=136-120\rho

Compute

The expression decreases with correlation, so choose the smallest feasible correlation.

\rho=-1\quad\Longrightarrow\quad \operatorname{Var}(Z)=136+120=256

Answer

The largest possible variance is 256.

\boxed{256\quad\text{(E)}}

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How to solve this Variance of Linear Combinations question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: maximum variance under unknown dependence

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables