Worked solution · Exam P

SOA Exam P Sample Question #481 Solution

This Exam P sample reference tests Discrete Uniform Distribution. For the equally likely integers 1 through 5, E[N]=3 and E[N²]=11. Linearity of expectation then gives E[N²+N+1]=11+3+1=15, so choice E 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 Discrete Uniform Distribution question

Setup

Each of the five possible duration values has probability one-fifth. The expected cost can be assembled from the first two moments of N.

\Pr(N=n)=\frac15,\qquad n=1,2,3,4,5

\operatorname{E}[N^2+N+1]=\operatorname{E}[N^2]+\operatorname{E}[N]+1

Model

Compute the first moment from the midpoint and the second moment from the five squares.

\operatorname{E}[N]=\frac{1+2+3+4+5}{5}=3

\operatorname{E}[N^2]=\frac{1^2+2^2+3^2+4^2+5^2}{5}=\frac{55}{5}=11

Compute

Substitute the two moments into the cost expectation.

\operatorname{E}[N^2+N+1]=11+3+1=15

Answer

The expected cost is 15.

\boxed{15\quad\text{(E)}}

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 is E[N²] = 11 alone; it omits both the linear term E[N] = 3 and the constant 1.
BThis computes E[N²] + 1 = 11 + 1 = 12 and drops the linear N term from the cost.
CThis evaluates the cost only at the middle duration, 3² + 3 + 1 = 13. In general, E[g(N)] is not obtained by replacing N with its mean.
DThis computes E[N²] + E[N] = 11 + 3 = 14 but omits the constant 1, whose expectation is also 1.

Original practice · fully worked

Original variant: score from the larger spinner result

Two fair four-sector spinners are operated independently, and each returns one of 1, 2, 3, or 4. Let M be the larger of the two results. A game awards 2M+1 points. Calculate the expected award.

A
3.125
B
6.000
C
6.250
D
7.250
E
9.000

Answer and calculation

The maximum has mean 1+15/16+12/16+7/16=25/8. Therefore the expected award is 2(25/8)+1=29/4=7.25, selecting choice D.

Setup

Use tail probabilities for the nonnegative integer maximum M.

\operatorname{E}[M]=\sum_{k=1}^{4}\Pr(M\ge k)

Model

The event M<k means both independent spinners fall below k.

\Pr(M\ge k)=1-\left(\frac{k-1}{4}\right)^2

Compute

Sum the four tail probabilities and then apply linearity to the award.

\operatorname{E}[M]=1+\frac{15}{16}+\frac{12}{16}+\frac{7}{16}=\frac{50}{16}=\frac{25}{8}

\operatorname{E}[2M+1]=2\left(\frac{25}{8}\right)+1=\frac{29}{4}=7.25

Answer

The expected award is 7.25 points.

\boxed{7.250\quad\text{(D)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

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More Exam P Univariate Random Variables practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Discrete Uniform Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: score from the larger spinner result

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables