Verified solution · Exam P

SOA Exam P Sample Question #358 Solution

This Exam P sample reference tests Poisson Distribution. This asks for the most probable count of a Poisson variable with mean 2.5. Adjacent probability ratios show that the mass increases through count 2 and decreases at count 3, so the unique mode is 2 and choice C.

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.

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Independent solution

How to solve this Poisson Distribution question

Setup

Write the ratio between consecutive Poisson masses.

\frac{\Pr(N=k)}{\Pr(N=k-1)}=\frac{2.5}{k}

Model

The mass function rises while this ratio exceeds one and falls after it drops below one.

\frac{\Pr(N=2)}{\Pr(N=1)}=1.25>1

\frac{\Pr(N=3)}{\Pr(N=2)}=\frac56<1

Compute

The direction changes after count two, identifying the peak.

\Pr(N=1)<\Pr(N=2)>\Pr(N=3)

\operatorname{mode}(N)=2

Answer

The modal count is two.

\boxed{2\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.

AThe value 1 can result from subtracting one standard deviation from the mean and rounding, but that quantity does not locate a Poisson mode.
BThe value 1.5 subtracts one from the mean. The Poisson mode for a noninteger mean is its integer floor, not the mean minus one.
DThe value 2.5 is the mean of the distribution. A count must be an integer, and the mean need not be the mode.
EThe value 3 rounds or takes the ceiling of the mean. The adjacent-mass ratio is already below one at count three.

Original practice · fully worked

Original variant: modal median of three calibration readings

Three independent calibration panels each display an integer selected uniformly from 1 through 5. The controller records M, the median of the three displayed values. Calculate the mode of M.

A
1 display unit
B
2 display units
C
3 display units
D
4 display units
E
5 display units

Variant answer in brief

The median CDF at m is the probability that at least two of three panels are at most m. Differencing it gives masses 13, 31, 37, 31, and 13 out of 125, so the unique mode is 3 and choice C.

Setup

Let q=m/5 be the probability that one panel displays at most m.

q=\Pr(X_i\le m)=\frac m5

Model

The median is at most m exactly when at least two of the three panels are at most m.

\Pr(M\le m)=3q^2(1-q)+q^3=3q^2-2q^3

Compute

Difference consecutive CDF values and compare the five masses.

\Pr(M=1,2,3,4,5)=\frac1{125}(13,31,37,31,13)

\operatorname{mode}(M)=3

Answer

The controller most often records three.

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

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #358 — 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: modal median of three calibration readings

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution