Worked solution · Exam P

SOA Exam P Sample Question #372 Solution

This Exam P sample reference tests Exponential Distribution. For a normal or continuous uniform distribution, symmetry makes the median equal to the mean. For an exponential distribution, the median is the mean multiplied by the constant natural logarithm of two. In every case, doubling the mean therefore doubles the median, so choice A is correct.

Independent analysis by ActuaryProof · reviewed July 31, 2026

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Independent solution

How to solve this Exponential Distribution question

Setup

Write the median-to-mean relation for each of the three distribution families.

m_{\mathrm{exp}}=(\ln 2)\mu

m_{\mathrm{normal}}=\mu

m_{\mathrm{uniform}}=\frac{a+b}{2}=\mu

Model

Each median is a fixed multiple of its distribution's mean.

m=c\mu,\qquad c\in\{\ln2,1\}

Compute

Replace the mean by twice its former value in the common relation.

m_{\mathrm{new}}=c(2\mu)=2(c\mu)=2m_{\mathrm{old}}

Answer

The doubling property holds for the exponential, normal, and continuous uniform distributions.

\boxed{\text{all three}\quad\text{(A)}}

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.

BThis excludes the normal family. A normal distribution is symmetric about its mean, so its median equals its mean and scales with it.
CThis excludes the continuous uniform family. Its midpoint is both mean and median, so the two quantities change together.
DThis excludes the exponential family. Its median is the natural logarithm of two times its mean, a fixed proportional relationship.
EAt least the normal and continuous uniform families qualify by symmetry, and the exponential family qualifies by its fixed median-to-mean ratio.

Original practice · fully worked

Original variant: median resale credit from wear grades

A machine's wear grade G is 1, 4, 7, or 10 with probabilities 0.20, 0.35, 0.25, and 0.20, respectively. Its resale credit is C=100/(G+1) dollars. Calculate the median resale credit.

A
4.000
B
9.091
C
15.748
D
20.000
E
25.000

Answer and calculation

The cumulative probability of the wear grade first exceeds one half at G=4, so 4 is its unique median. The decreasing credit rule carries that median to 100/(4+1)=20 dollars, and direct ordering of the credit masses confirms choice D.

Setup

Locate where the cumulative wear-grade probability first reaches one half.

\Pr(G\le1)=0.20,\qquad \Pr(G\le4)=0.20+0.35=0.55

m_G=4

Model

The credit is a strictly decreasing function of grade. A unique median maps to the corresponding transformed value even though the ordering reverses.

m_C=\frac{100}{m_G+1}

Compute

Apply the resale-credit rule to grade four.

m_C=\frac{100}{4+1}=20

Answer

The median resale credit is 20 dollars.

\boxed{20.000\quad\text{(D)}}

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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: median resale credit from wear grades

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables