Worked solution · Exam P

SOA Exam P Sample Question #171 Solution

This Exam P sample reference tests Variance. Let b=P(XY=1), so P(XY=2)=3b and P(XY=0)=1-4b. The variance is 13b-49b², maximized at b=13/98; therefore P(XY=0)=23/49, choice C.

Independent analysis by ActuaryProof · reviewed July 31, 2026

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Independent solution

How to solve this Variance question

Setup

Collapse the joint table to the distribution of Z=XY.

\Pr(Z=1)=b,\quad \Pr(Z=2)=3b

\Pr(Z=0)=1-4b

Model

Compute the first two moments as functions of the single free probability.

E[Z]=7b

E[Z^2]=13b

\operatorname{Var}(Z)=13b-49b^2

Compute

Set the derivative of the concave quadratic to zero.

\frac{d}{db}\operatorname{Var}(Z)=13-98b=0

b=\frac{13}{98}

\Pr(Z=0)=1-4b=\frac{23}{49}

Answer

The event that either factor is zero is exactly the event Z=0.

\boxed{\frac{23}{49}\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.

AIf P(Z=0)=11/25, normalization would give b=0.14. The variance derivative there is 13-98(0.14)=-0.72, so this is not the maximizer.
BThe value 23/50 is a premature decimal-style denominator change. The exact optimizer b=13/98 gives 1-4b=23/49, not 23/50.
DThe value 26/49 is 1-23/49=P(Z>0). It is the complement of the requested zero-product event.
EIf P(Z=0)=14/25, then b=0.11 and the derivative is 2.22 rather than zero; the variance is still increasing there.

Original practice · fully worked

Original variant: minimize a blended estimate's variance

Two unbiased estimates X and Y target the same calibration constant. Their variances are 9 and 4, and their covariance is 1. For W_a=aX+(1-a)Y, determine the real value of a that minimizes Var(W_a).

A
0
B
7/24
C
3/11
D
4/13
E
1/3

Answer and calculation

Expanding the weighted-sum variance gives 11a²-6a+4. Its derivative vanishes at a=3/11, and the positive quadratic coefficient makes this the minimum, choice C.

Setup

Apply the variance formula to the weighted combination.

\operatorname{Var}(W_a)=a^2\operatorname{Var}(X)+(1-a)^2\operatorname{Var}(Y)+2a(1-a)\operatorname{Cov}(X,Y)

Model

Substitute the three supplied second-moment quantities and collect powers of a.

\operatorname{Var}(W_a)=9a^2+4(1-a)^2+2a(1-a)

\operatorname{Var}(W_a)=11a^2-6a+4

Compute

Differentiate the convex quadratic and solve its first-order condition.

\frac{d}{da}\operatorname{Var}(W_a)=22a-6=0

a=\frac{3}{11}

Answer

The minimum-variance weight on X is 3/11.

\boxed{\frac{3}{11}\quad\text{(C)}}

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Next steps for Exam P Multivariate Random Variables

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How to solve this Variance question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: minimize a blended estimate's variance

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables