Verified solution · Exam P

SOA Exam P Sample Question #176 Solution

This Exam P sample reference tests Convolution. Convolving the two claim-count distributions gives total-count probabilities 0.10, 0.21, 0.33, 0.23, 0.11, and 0.02 for totals 0 through 5. The largest mass is 0.33 at total 2, choice C.

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 Convolution question

Setup

Let G and B be the independent individual claim counts and T=G+B.

\Pr(T=t)=\sum_g\Pr(G=g)\Pr(B=t-g)

Model

Compute the convolution masses for the central candidate totals.

\Pr(T=1)=0.5(0.3)+0.3(0.2)=0.21

\Pr(T=2)=0.5(0.4)+0.3(0.3)+0.2(0.2)=0.33

\Pr(T=3)=0.5(0.1)+0.3(0.4)+0.2(0.3)=0.23

Compute

Complete the distribution to verify the global maximum.

(\Pr(T=0),\ldots,\Pr(T=5))=(0.10,0.21,0.33,0.23,0.11,0.02)

\max_t\Pr(T=t)=0.33=\Pr(T=2)

Answer

The total-count mode is 2.

\boxed{2\quad\text{(C)}}

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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.

AAt total 0 the convolution mass is 0.5(0.2)=0.10, which is below the mass at 2.
BAt total 1 the two feasible products sum to 0.21, still below the maximum 0.33.
DAt total 3 the feasible products sum to 0.23. It is the second-largest mass, not the mode.
EAt total 4 the mass is 0.3(0.1)+0.2(0.4)=0.11, well below 0.33.

Original practice · fully worked

Original variant: convolve two continuous delays

A calibration delay X is uniform on the interval from 0 to 2 minutes. An independent network delay Y is exponential with rate 1 per minute. For the total delay S=X+Y, calculate the density f_S(1).

A
0.18394
B
0.31606
C
0.50000
D
0.63212
E
0.85914

Variant answer in brief

At total 1, the uniform component ranges from 0 to 1. Convolution gives one half times the integral of exp(−(1−x)) over that interval, or (1-exp(−1))/2=0.31606, choice B.

Setup

Write the two component densities on their supports.

f_X(x)=\frac12\quad(0\le x\le2)

f_Y(y)=e^{-y}\quad(y\ge0)

Model

For a total of one, both supports are satisfied precisely when the uniform delay lies between zero and one.

f_S(1)=\int_0^1 f_X(x)f_Y(1-x)\,dx

Compute

Evaluate the convolution integral.

f_S(1)=\frac12\int_0^1e^{-(1-x)}\,dx

f_S(1)=\frac{1-e^{-1}}{2}=0.3160603

Answer

The total-delay density at one minute is approximately 0.31606.

\boxed{0.31606\quad\text{(B)}}

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ActuaryProof. “SOA Exam P Sample Question #176 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Convolution question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: convolve two continuous delays

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution