Worked solution · Exam P

SOA Exam P Sample Question #425 Solution

This Exam P sample reference tests Poisson Distribution. The zero-count probability gives λ=-ln(0.18)=1.7148. A Poisson distribution with a noninteger mean has unique mode floor(λ)=1, so 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.

Read the solution Open official source

Independent solution

How to solve this Poisson Distribution question

Setup

Use the Poisson probability at zero to recover the mean parameter.

\Pr(X=0)=e^{-\lambda}=0.18

Model

Take logarithms and identify the interval containing the parameter.

\lambda=-\ln(0.18)=1.7147984281\ldots

1<\lambda<2

Compute

For a noninteger Poisson mean, the unique modal count is the greatest integer below the mean.

\operatorname{mode}(X)=\lfloor\lambda\rfloor=1

Answer

The most likely count is 1.

\boxed{1\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.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

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

AAlthough Pr(X=0)=0.18, Pr(X=1)=0.18lambda=0.3087 is larger, so zero is not the mode.
CThis rounds the mean 1.7148 to 2. In fact, Pr(X=2)/Pr(X=1)=λ/2=0.8574<1, so the mass has already begun decreasing.
DThe value 5 is floor(1/0.18), but the reciprocal of the zero-count probability is not the Poisson mode.
EThe value 6 is 1/0.18 rounded to the nearest integer; that reciprocal has no modal-count interpretation.

Original practice · fully worked

Original variant: most likely sensor delay

A sensor's post-maintenance delay T has a continuous density proportional to t²(8-t) for 0<t<8 and zero elsewhere. Calculate the mode of T.

A
0
B
4.000
C
4.800
D
5.333
E
8.000

Answer and calculation

The normalizing constant does not affect the location of the maximum. Differentiating t²(8-t) gives t(16-3t), whose interior maximum is t=16/3=5.333 and choice D.

Setup

Ignore the positive normalizing constant and maximize the density kernel on its support.

g(t)=t^2(8-t),\qquad 0<t<8

Model

Differentiate the kernel to locate its stationary points.

g'(t)=16t-3t^2=t(16-3t)

Compute

The interior critical point is 16/3. The derivative is positive before it and negative after it.

t=\frac{16}{3}=5.333333\ldots

Answer

The density reaches its maximum at 16/3.

\boxed{5.333\quad\text{(D)}}

Keep practicing

Next steps for Exam P Univariate Random Variables

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

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

How to solve this Poisson Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: most likely sensor delay

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables