Worked solution · Exam P

SOA Exam P Sample Question #553 Solution

This Exam P sample reference tests Exponential Distribution. Writing the exponential percentile through its survival probability identifies the scale parameter as 4. The median then solves exp(-m/4)=0.5, so m=4 ln 2 and choice B is correct.

Independent analysis by ActuaryProof · reviewed July 30, 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

Let beta be the exponential mean and write the survival function in terms of that scale.

S(x)=e^{-x/\beta}

Model

At the supplied 40th percentile, the survival probability is 0.60. Substitute the given percentile value into the survival equation.

e^{-4\ln(5/3)/\beta}=0.60=\frac35

Compute

Because three-fifths equals exp[-ln(5/3)], the scale is 4. Apply the median survival probability to that scale.

\beta=4

e^{-m/4}=0.50

m=-4\ln(0.50)=4\ln2

Answer

The exponential median is four times the natural logarithm of two.

\boxed{4\ln2\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.

AThe expression 5ln(5/3) linearly scales the supplied 40th-percentile value by 0.50/0.40. Exponential quantiles scale with -ln(1-p), not directly with p.
CThe value 5ln(25/12)=3.66985 has F(x)=1-exp(-x/4)=0.60047 after the scale is calibrated; it effectively targets about the 60th percentile rather than the median.
DThe value 4 is the inferred exponential mean beta. The median is beta ln 2, which is smaller than the mean.
EThe value 5 drops the logarithmic percentile conversion and reports a number from the survival ratio. With beta=4, F(5)=0.71350, not 0.50.

Original practice · fully worked

Original variant: extra repair time after escalation

A device's repair time is exponentially distributed with a median of 6 minutes. A case is escalated when its repair time passes the 75th-percentile threshold. Given that a case has just been escalated, calculate the probability that its repair takes at least 4 additional minutes.

A
0.1575
B
0.2500
C
0.5000
D
0.6300
E
0.7500

Answer and calculation

Memorylessness removes the elapsed threshold from the conditional probability. A median of 6 gives four-minute survival 2⁽⁻⁴⁄⁶⁾=0.6300, so choice D is correct.

Setup

Convert the six-minute median into an exponential rate.

e^{-6\lambda}=0.50

\lambda=\frac{\ln2}{6}

Model

Let t denote the escalation threshold. Exponential memorylessness makes the extra four-minute survival independent of t.

\Pr(T>t+4\mid T>t)=\Pr(T>4)

Compute

Evaluate four-minute survival using the rate obtained from the median.

\Pr(T>4)=e^{-4\ln2/6}=2^{-2/3}=0.6299605249

Answer

The conditional probability is approximately 0.6300.

\boxed{0.6300\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 Exponential Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: extra repair time after escalation

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables