Worked 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 July 31, 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 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)}}

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.

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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 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

Answer and calculation

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

Next steps for Exam P Univariate Random Variables

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How to solve this Exponential Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: decreasing exponential dashboard score

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables