Worked solution · Exam P

SOA Exam P Sample Question #390 Solution

This Exam P sample reference tests Gamma Distribution. The mean and variance identify gamma shape 2 and scale 3. Its survival probability at 4 is exp(-4/3)(1+4/3)=0.615060, which rounds to 0.62 and choice E.

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 Gamma Distribution question

Setup

Write the gamma moment equations in shape-scale form.

\alpha\theta=6,\qquad \alpha\theta^2=18

Model

Divide the variance by the mean to obtain the scale, then recover the shape.

\theta=\frac{18}{6}=3,\qquad \alpha=\frac{6}{3}=2

Compute

Use the closed-form survival function for an integer shape of two.

\Pr(X>4)=e^{-4/3}\left(1+\frac43\right)

\Pr(X>4)=\frac73e^{-4/3}=0.6150599889\ldots

Answer

The upper-tail probability rounds to 0.62.

\boxed{0.62\quad\text{(E)}}

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.

AFor shape 2, a survival probability of 0.54 would require scale approximately 2.574 rather than 3. This reflects an understated scale that fails the supplied moment ratio Var(X)/E[X]=3.
BThe value 0.56 corresponds to a shape-2 scale near 2.678. It is a nearby table value for an understated scale and does not satisfy both supplied gamma moment equations.
CThe value 0.58 corresponds to a shape-2 scale near 2.789. It is a nearby table value for the wrong parameter, not the distribution with scale 3.
DThe value 0.60 corresponds to a shape-2 scale near 2.906. Rounding the scale downward before evaluating the tail changes the result; the exact moment ratio gives scale 3.

Original practice · fully worked

Original variant: compare two gamma completion times

Two independent processing times X and Y have gamma distributions with a common scale of 4. Their shape parameters are 2 and 3, respectively. Calculate the probability that X exceeds Y.

A
0.1250
B
0.1875
C
0.3125
D
0.4000
E
0.6875

Answer and calculation

For independent gamma variables with a common scale, U=X/(X+Y) is Beta(2,3). Thus P(X>Y)=P(U>1/2)=5/16=0.3125, choice C.

Setup

Transform the two common-scale gamma variables to their share of the total.

U=\frac{X}{X+Y}\sim\operatorname{Beta}(2,3)

Model

The event X greater than Y is equivalent to the share U exceeding one half.

f_U(u)=12u(1-u)^2,\qquad 0<u<1

\{X>Y\}=\left\{U>\frac12\right\}

Compute

Integrate the beta density over the upper half of its support.

\Pr(X>Y)=\int_{1/2}^{1}12u(1-u)^2\,du

=1-\left[6\left(\frac12\right)^2-8\left(\frac12\right)^3+3\left(\frac12\right)^4\right]=\frac5{16}=0.3125

Answer

The probability that X finishes later is 0.3125.

\boxed{0.3125\quad\text{(C)}}

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Next steps for Exam P Univariate Random Variables

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How to solve this Gamma Distribution question

Setup

Model

Compute

Answer

Probability Univariate Distributions Practice Workbook

Why the other letters tempt

Original variant: compare two gamma completion times

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Univariate Random Variables