Worked solution · Exam P

SOA Exam P Sample Question #294 Solution

This Exam P sample reference tests Expected Value. Weight each percentage change by its economic-state probability. Both stocks have mean change 0.068, but their variances are 0.011676 and 0.005296, respectively. Thus Stock X has the larger variance while the means are equal, which is choice B.

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 Expected Value question

Setup

Represent the three state probabilities and the two decimal price-change vectors.

\mathbf p=(0.30,0.50,0.20)

\mathbf x=(0.18,0.08,-0.13),\qquad \mathbf y=(0.15,0.07,-0.06)

Model

Compute the two probability-weighted means.

E[X]=0.30(0.18)+0.50(0.08)+0.20(-0.13)=0.068

E[Y]=0.30(0.15)+0.50(0.07)+0.20(-0.06)=0.068

Compute

Weight the squared deviations from the common mean.

\operatorname{Var}(X)=\sum_i p_i(x_i-0.068)^2=0.011676

\operatorname{Var}(Y)=\sum_i p_i(y_i-0.068)^2=0.005296

Answer

The means are equal, and Stock X has the larger variance.

\boxed{E[X]=E[Y],\quad \operatorname{Var}(X)>\operatorname{Var}(Y)\quad\text{(B)}}

Test the method on unseen problems

Probability Univariate Distributions Practice Workbook

30 original univariate 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.

AThis correctly identifies the larger variance of Stock X but misstates the means. Direct weighting gives 0.068 for each stock.
CThis also identifies the variance ordering correctly, but Stock X's mean is not smaller; the two weighted means are equal.
DNeither part is supported: the variances differ by 0.00638, and Stock X does not have a larger mean.
EThe mean comparison is correct, but equal means do not imply equal variances. Stock X's more dispersed state changes produce the larger variance.

Original practice · fully worked

Original variant: mirrored calibration indexes

A calibration score R has mean 4 and variance 9. Two reporting indexes are defined by U=2R-5 and V=-2R+11. Determine which statement about the means and variances of U and V is true.

A
Both E[U] and Var(U) exceed their V counterparts.
B
E[U] = E[V], but Var(U) > Var(V).
C
Var(U) = Var(V), but E[U] < E[V].
D
E[U] = E[V] and Var(U) = Var(V).
E
Var(U) = Var(V), but E[U] > E[V].

Answer and calculation

Both affine transformations have mean 3. Their slopes have equal magnitude, so both variances are 4 times Var(R), or 36. The means and variances are therefore equal, making choice D correct.

Setup

Apply linearity of expectation to the two reporting indexes.

E[U]=2E[R]-5=2(4)-5=3

E[V]=-2E[R]+11=-2(4)+11=3

Model

An additive constant does not affect variance, while a multiplicative coefficient is squared.

\operatorname{Var}(U)=2^2\operatorname{Var}(R)

\operatorname{Var}(V)=(-2)^2\operatorname{Var}(R)

Compute

Evaluate the two transformed variances.

\operatorname{Var}(U)=4(9)=36

\operatorname{Var}(V)=4(9)=36

Answer

Both pairs of moments are equal.

\boxed{E[U]=E[V]=3,\quad \operatorname{Var}(U)=\operatorname{Var}(V)=36\quad\text{(D)}}

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How to solve this Expected Value question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: mirrored calibration indexes

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables