Worked solution · Exam P

SOA Exam P Sample Question #229 Solution

This Exam P sample reference tests Uniform Distribution. The supplemental payment has a zero region, a linear region, and a capped region. Its first two moments are 275 and 116666.67, producing variance 41041.67, choice C.

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 Uniform Distribution question

Setup

Write the supplemental benefit as a limited excess payment applied to the underlying expense X.

Y=\min\left((X-400)_+,500\right)

f_X(x)=\frac{1}{1000},\qquad 200\le x\le 1200

Model

Split the moment calculations at the attachment point 400 and the exhaustion point 900.

\operatorname{E}[Y]=\frac{1}{1000}\int_{400}^{900}(x-400)\,dx+500\Pr(X>900)

\operatorname{E}[Y^2]=\frac{1}{1000}\int_{400}^{900}(x-400)^2\,dx+500^2\Pr(X>900)

Compute

Evaluate the continuous payment region and add the probability mass at the limit.

\Pr(X>900)=0.30

\operatorname{E}[Y]=125+150=275

\operatorname{E}[Y^2]=\frac{500^3}{3(1000)}+500^2(0.30)=116666.6667

\operatorname{Var}(Y)=116666.6667-275^2=41041.6667

Answer

The variance rounds to 41,042.

\boxed{41042\quad\text{(C)}}

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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.

AThe value 20,833 is 500 squared divided by 12, the variance of a uniform variable on [0,500]. The payment is not uniform because it also has atoms at zero and 500.
BThe value 26,042 results from integrating only the uncapped linear segment: 41666.67-125 squared=26041.67. It drops the 30% probability mass at the 500 limit.
DThe value 53,333 is 800 squared divided by 12. It treats the uncapped remaining expense as uniform on [0,800] and ignores both the zero-payment and cap mechanics.
EThe value 83,333 is the variance of the original uniform expense, 1000 squared divided by 12. It does not transform the expense through either plan.

Original practice · fully worked

Original variant: variance under a franchise trigger

A drone repair bill X is uniformly distributed from 0 to 10 thousand credits. A grant pays the entire bill when X exceeds 6 and pays zero otherwise. Let Y be the grant payment. Calculate Var(Y), in squared thousand-credit units.

A
3.200
B
8.333
C
10.240
D
15.893
E
26.133

Answer and calculation

The grant has mean integral from 6 to 10 of x/10, equal to 3.2, and second moment 26.1333. Its variance is 26.1333-3.2 squared=15.8933, choice D.

Setup

Express the all-or-nothing grant payment created by the franchise trigger.

Y=X\mathbf{1}_{\{X>6\}}

f_X(x)=\frac1{10},\qquad 0<x<10

Model

Integrate the first two payment moments over bills that pass the trigger.

\operatorname{E}[Y]=\frac1{10}\int_6^{10}x\,dx=3.2

\operatorname{E}[Y^2]=\frac1{10}\int_6^{10}x^2\,dx=\frac{392}{15}

Compute

Subtract the squared mean from the second raw moment.

\operatorname{Var}(Y)=\frac{392}{15}-3.2^2

\operatorname{Var}(Y)=\frac{1192}{75}=15.8933333

Answer

The grant-payment variance is approximately 15.893.

\boxed{15.893\quad\text{(D)}}

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How to solve this Uniform Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: variance under a franchise trigger

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables