Worked solution · Exam P

SOA Exam P Sample Question #560 Solution

This Exam P sample reference tests Exponential Distribution. Partial nonreimbursement occurs when a loss exceeds the applicable cap. Dividing the two exponential tail probabilities cancels the unknown base cap and leaves exp(-2/6)=exp(-1/3), so 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.

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Independent solution

How to solve this Exponential Distribution question

Setup

A capped reimbursement leaves part of a loss unpaid exactly when the loss exceeds the cap.

p_A=\Pr(X>m),\qquad p_B=\Pr(X>m+2)

Model

Apply the survival function for an exponential variable with mean six.

p_A=e^{-m/6},\qquad p_B=e^{-(m+2)/6}

Compute

Form the requested ratio and cancel the common cap term.

\frac{p_B}{p_A}=\frac{e^{-(m+2)/6}}{e^{-m/6}}

\frac{p_B}{p_A}=e^{-2/6}=e^{-1/3}

Answer

The ratio is exp(-1/3), approximately 0.7165.

\boxed{e^{-1/3}\quad\text{(B)}}

Test the method on unseen problems

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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.

AThis multiplies the cap increase by the mean, producing exp(−6(2))=exp(−12), instead of dividing the increase by the mean.
CThis divides the two probabilities of full reimbursement, (1-exp(−(m+2)/6))/(1-exp(−m/6)), rather than the two unpaid-tail probabilities.
DThis both uses full-reimbursement probabilities and treats six as an exponential rate, producing exponents -6(m+2) and -6m.
EThis uses the linear cap ratio m/(m+2), even though exponential tail probabilities change multiplicatively rather than inversely with the cap.

Original practice · fully worked

Original variant: expected capped incidents

The number of repair incidents during a season is Poisson with mean 4. Incident costs are independent exponential random variables with mean 6 units and are independent of the count. A contract reimburses at most 8 units per incident. Calculate the expected number of incidents for which some cost remains unreimbursed.

A
0.2636
B
1.0544
C
2.9456
D
4.0000
E
8.0000

Answer and calculation

An incident is only partly reimbursed with probability exp(-8/6)=0.2636. Multiplying by the expected incident count gives 4exp(-8/6)=1.0544, so choice B is correct.

Setup

Identify the event that an incident cost exceeds the reimbursement cap.

q=\Pr(X>8)=e^{-8/6}=e^{-4/3}

Model

Conditional on N incidents, the expected number exceeding the cap is Nq. Average this conditional mean over N.

E[K\mid N]=Nq

E[K]=E[N]q

Compute

Use the Poisson mean and the exponential tail probability.

E[K]=4e^{-4/3}=1.054388552

Answer

The expected number of partially unreimbursed incidents is approximately 1.0544.

\boxed{1.0544\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: expected capped incidents

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables