Verified solution · Exam P

SOA Exam P Sample Question #507 Solution

This Exam P sample reference tests Gamma Distribution. The density kernel is gamma with shape 6 and rate 0.01. Its mean is 6/0.01=600, so choice E is correct.

Independent analysis by ActuaryProof · reviewed August 4, 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 Gamma Distribution question

Setup

Match the power and exponential factor to a gamma density written in shape-rate form.

x^5e^{-0.01x}=x^{6-1}e^{-0.01x}

\alpha=6,\qquad \beta=0.01

Model

For a gamma variable with shape alpha and rate beta, the normalizing constant does not need to be found separately to use the mean formula.

\operatorname{E}[X]=\frac{\alpha}{\beta}

Compute

Substitute the identified shape and rate.

\operatorname{E}[X]=\frac{6}{0.01}=600

Answer

The expected value is 600.

\boxed{600\quad\text{(E)}}

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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 reports only the gamma scale 1/0.01=100 and omits the shape multiplier 6.
BThis treats the polynomial factor as x rather than x⁵, incorrectly assigning shape 2 and obtaining 2/0.01=200.
CThis halves the identified shape from 6 to 3 and therefore computes 3/0.01=300.
DThis reports the gamma mode (alpha-1)/beta=(6-1)/0.01=500 instead of the mean.

Original practice · fully worked

Original variant: infer gamma variance from field summaries

A sensor's recalibration delay T follows a gamma distribution with shape alpha and scale θ. Field engineers report a mode of 6 minutes and a coefficient of variation of 1/2. Calculate Var(T).

A
4
B
8
C
12
D
16
E
64

Variant answer in brief

The coefficient of variation gives shape 4, and the mode then gives scale 2. Thus the variance is 4(2²)=16, so choice D is correct.

Setup

Use the gamma coefficient of variation to recover the shape parameter.

\operatorname{CV}(T)=\frac{1}{\sqrt{\alpha}}=\frac12

\alpha=4

Model

For shape greater than one, the gamma mode identifies the scale after the shape is known.

\operatorname{mode}(T)=(\alpha-1)\theta

Compute

Recover the scale and then evaluate the gamma variance.

6=(4-1)\theta\quad\Longrightarrow\quad\theta=2

\operatorname{Var}(T)=\alpha\theta^2=4(2^2)=16

Answer

The delay variance is 16 square minutes.

\boxed{16\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #507 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Gamma Distribution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: infer gamma variance from field summaries

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution