Verified solution · Exam P

SOA Exam P Sample Question #386 Solution

This Exam P sample reference tests Poisson Distribution. This problem expresses an increasing sequence of charges as a triangular-number function of a Poisson count. Using the first two Poisson moments gives an expected total of 12 and choice D.

Independent analysis by ActuaryProof · reviewed August 4, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

Read the solution Open official source

Independent solution

How to solve this Poisson Distribution question

Setup

Write the total charge incurred when the count equals n.

T(n)=1+2+\cdots+n=\frac{n(n+1)}2

T=\frac{N(N+1)}2

Model

Use the raw second moment of a Poisson variable with mean four.

E[N]=4,\qquad \operatorname{Var}(N)=4

E[N^2]=\operatorname{Var}(N)+E[N]^2=4+16=20

Compute

Apply linearity of expectation to the quadratic charge.

E[T]=\frac12\left(E[N^2]+E[N]\right)

E[T]=\frac12(20+4)=12

Answer

The expected total charge is 12.

\boxed{12\quad\text{(D)}}

Continue without hunting through PDFs

All 718 Exam P sample solutions in syllabus order

The searchable 3108-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam P study manual PDF

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis evaluates the unordered-pair count at the mean of four, giving six. It uses the wrong charge function and moves the expectation through a nonlinear calculation.
BThis takes half the second factorial moment, giving eight unordered pairs, but omits the additional linear term in the triangular charge.
CThis inserts the mean of four into the triangular formula and gets ten. A nonlinear function evaluated at the mean does not generally equal its expectation.
EThis counts the linear contribution twice, combining the second moment 20 with twice the mean to obtain 14.

Original practice · fully worked

Original variant: expected review cost for alert triples

A monitoring system records a Poisson number N of alerts per shift with mean 3. After each shift, every unordered group of three distinct alerts is reviewed, at a cost of 2 credits per group. Calculate the expected review cost.

A
4.5 credits
B
9 credits
C
18 credits
D
27 credits
E
54 credits

Variant answer in brief

The reviewed groups are the unordered triples of alerts. The third Poisson factorial moment gives an expected 4.5 groups, so the two-credit review cost has mean 9 and choice B.

Setup

Express the number of unordered three-alert groups as a function of the shift count.

G=\binom N3=\frac{N(N-1)(N-2)}{3!}

C=2G

Model

Use the third factorial moment of a Poisson variable.

E[N(N-1)(N-2)]=\lambda^3

\lambda=3

Compute

Convert ordered factorial triples to unordered groups and apply the cost per group.

E[G]=\frac{3^3}{3!}=\frac{27}{6}=4.5

E[C]=2(4.5)=9

Answer

The expected review cost is 9 credits per shift.

\boxed{9\ \text{credits}\quad\text{(B)}}

Keep practicing

Turn this solution into a complete study sequence

The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

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

Cite this solution

ActuaryProof. “SOA Exam P Sample Question #386 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Poisson Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: expected review cost for alert triples

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution