Worked solution · Exam P

SOA Exam P Sample Question #204 Solution

This Exam P sample reference tests Exponential Distribution. Equating the two exponential interval probabilities gives exp(-r/100)=exp(-0.6)-exp(-0.4)+exp(-0.5). Taking logarithms yields r=72.3561, which selects choice D.

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

For mean 100, use the exponential survival function rather than treating equal probabilities as equal interval lengths.

S(x)=e^{-x/100}

Model

An interval probability is the difference of survival values at its endpoints.

S(40)-S(50)=S(60)-S(r)

Compute

Isolate the unknown survival value and invert it.

e^{-r/100}=e^{-0.6}-e^{-0.4}+e^{-0.5}=0.485003

r=-100\ln(0.485003)=72.3560513

Answer

The matching listed endpoint is 72.36.

\boxed{72.36\quad\text{(D)}}

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.

AAt r=68.26, the later interval probability is exp(−0.6)-exp(−0.6826)=0.043510, not the required 0.063789. This can result from taking logs before completing the tail subtraction.
BThe value 70.00 assigns the later interval the same width of 10. Its probability is 0.052226, smaller because the exponential density has declined.
CAt r=70.51, the later interval probability is 0.054752, still below 0.063789. Rounding or linearly interpolating cdf values does not solve the exponential equation.
EThe endpoint 75.00 gives probability exp(−0.6)-exp(−0.75)=0.076445, which overshoots the required interval probability.

Original practice · fully worked

Original variant: exponential percentile conversion

A cloud task has an exponential completion time. Its 60th percentile is 8 minutes. Calculate the 90th percentile of the completion time.

A
1.650
B
8.000
C
12.000
D
18.421
E
20.104

Answer and calculation

The survival probabilities at the two percentiles are 0.40 and 0.10. Exponential scaling gives x=8 ln(0.10)/ln(0.40)=20.1035 minutes, choice E.

Setup

Convert the percentile statements to survival probabilities.

S(8)=0.40

S(x_{0.90})=0.10

Model

For an exponential variable, the logarithm of survival is linear in time.

e^{-8\lambda}=0.40

e^{-\lambda x_{0.90}}=0.10

Compute

Divide the two logarithmic equations to eliminate the unknown rate.

\frac{x_{0.90}}{8}=\frac{\ln(0.10)}{\ln(0.40)}

x_{0.90}=20.1035328

Answer

The 90th percentile is approximately 20.104 minutes.

\boxed{20.104\quad\text{(E)}}

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: exponential percentile conversion

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables