SOA Exam P Sample Question #655. Official problem wording is withheld; follow the source link to read it.

Verified solution · Exam P

SOA Exam P Sample Question #655

Scaling the constant Poisson rate from thirty years to ten gives mean and variance 2/3. The standard deviation is the square root of 2/3, approximately 0.816, so the answer is D.

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

Answer in brief

Scaling the constant Poisson rate from thirty years to ten gives mean and variance 2/3. The standard deviation is the square root of 2/3, approximately 0.816, so the answer is D.

Setup

Convert the stated long-window count rate into the requested time window.

\lambda_{10}=2\left(\frac{10}{30}\right)=\frac23

Model

A Poisson count has variance equal to its mean.

N_{10}\sim\operatorname{Poisson}\left(\frac23\right)

\operatorname{Var}(N_{10})=\frac23

Compute

Take the square root of the variance to obtain the standard deviation.

\operatorname{SD}(N_{10})=\sqrt{\frac23}=0.8164965809\ldots

Answer

The requested standard deviation rounds to 0.816.

\boxed{0.816\quad\text{(D)}}

Error diagnosis

Why the other letters tempt

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

AThis divides the correct standard deviation by the square root of ten, as though averaging ten observations rather than counting events over ten years.
BThis corresponds to using variance 2/9, which scales the thirty-year parameter by an extra factor of one third.
CThis is the ten-year Poisson mean and variance 2/3, not their square root.
EThis is the square root of the full thirty-year mean 2 and ignores the shorter time window.

Original practice · fully worked

Original variant: priority server alerts

Priority alerts arrive at a server as a Poisson process with an average of 3.6 alerts per twelve-hour monitoring window. Calculate the standard deviation of the number of alerts during the next five hours.

A
0.417
B
1.000
C
1.225
D
1.500
E
1.897

Variant answer in brief

The five-hour mean and variance are 3.6(5/12) = 1.5. The standard deviation is the square root of 1.5, approximately 1.225, so choice C.

Setup

Scale the twelve-hour rate to five hours.

\lambda_5=3.6\left(\frac{5}{12}\right)=1.5

Model

The five-hour alert count is Poisson, so its variance is also 1.5.

A_5\sim\operatorname{Poisson}(1.5)

\operatorname{Var}(A_5)=1.5

Compute

Take the square root of the five-hour variance.

\operatorname{SD}(A_5)=\sqrt{1.5}=1.224744871\ldots

Answer

The standard deviation rounds to 1.225.

\boxed{1.225\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #655 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.