Verified solution · Exam P

SOA Exam P Sample Question #428 Solution

This Exam P sample reference tests Exponential Distribution. For an exponential lifetime with mean 7.2, the lower-tail equation is 1-exp(-t/7.2)=0.05. Solving gives t=-7.2ln(0.95)=0.3693, so choice B.

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 Exponential Distribution question

Setup

Express the exponential distribution in terms of its given mean.

F(t)=1-e^{-t/7.2},\qquad t\ge0

Model

Set the cumulative probability at the unknown cutoff equal to the specified lower-tail probability.

1-e^{-t/7.2}=0.05

e^{-t/7.2}=0.95

Compute

Take logarithms and solve for the cutoff.

t=-7.2\ln(0.95)=0.3693117196\ldots

Answer

The cutoff is approximately 0.369 years.

\boxed{0.369\quad\text{(B)}}

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.

AThe value 0.007 is approximately 0.05/7.2, which divides by the mean instead of inverting the exponential CDF; its lower-tail probability is only 0.00097.
CAt t=0.416, the exponential lower-tail probability is 1-exp(-0.416/7.2)=0.05614 rather than 0.05.
DAt t=0.501, the lower-tail probability is 1-exp(-0.501/7.2)=0.06722, which overshoots the target.
EAt t=0.720, the lower-tail probability is 1-exp(-0.720/7.2)=0.09516, nearly twice the target.

Original practice · fully worked

Original variant: decreasing exponential dashboard score

An automated aging test records a duration X that is exponentially distributed with mean 6 hours. Instead of displaying X, the instrument reports the decreasing score Y=100 exp(-X/6). Calculate the 25th percentile of Y.

A
6.000
B
25.000
C
36.788
D
50.000
E
75.000

Variant answer in brief

For 0<y<100, the event Y≤y is the exponential survival event X≥-6ln(y/100), whose probability is y/100. Thus Y is uniform on (0,100), and its 25th percentile is 25, choice B.

Setup

Write the survival function of the recorded duration.

\Pr(X\ge x)=e^{-x/6},\qquad x\ge0

Model

Because the displayed score decreases with X, convert its lower-tail event to an upper-tail event for X.

\Pr(Y\le y)=\Pr\left(100e^{-X/6}\le y\right)

\Pr(Y\le y)=\Pr\left(X\ge-6\ln\frac{y}{100}\right)

Compute

Evaluate the exponential survival function, then set the resulting CDF equal to 0.25.

F_Y(y)=e^{-[-6\ln(y/100)]/6}=\frac{y}{100}

F_Y(q_{0.25})=0.25\quad\Longrightarrow\quad q_{0.25}=25

Answer

The 25th percentile of the displayed score is 25.

\boxed{25.000\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 #428 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Exponential Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: decreasing exponential dashboard score

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution