Worked solution · Exam P

SOA Exam P Sample Question #243 Solution

This Exam P sample reference tests Joint Distributions. The benefit is 0 only when both discrete risks are zero, 50 when their maximum is one, and 100 otherwise. Those probabilities are 1/25, 3/25, and 21/25, so the expected benefit is 90, choice B.

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 Joint Distributions question

Setup

Let M be the larger of the two independent discrete risk levels and express the capped benefit through M.

M=\max(X,Y),\qquad B=\min(50M,100)

Model

Since each ordered pair in the five-by-five grid is equally likely, count the pairs that place the maximum in each benefit tier.

\Pr(B=0)=\Pr(M=0)=\frac1{25}

\Pr(B=50)=\Pr(M=1)=\frac{2^2-1^2}{25}=\frac3{25}

\Pr(B=100)=1-\frac4{25}=\frac{21}{25}

Compute

Weight each benefit level by its probability.

\operatorname{E}[B]=0\left(\frac1{25}\right)+50\left(\frac3{25}\right)+100\left(\frac{21}{25}\right)=90

Answer

The expected capped benefit is 90.

\boxed{90\quad\text{(B)}}

Test the method on unseen problems

Probability Multivariate & Risk Models Practice Workbook

30 original multivariate 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 70 is obtained by applying the capped schedule to just one uniform risk level: (0+50+100+100+100)/5=70. The contract depends on the larger of two levels.
CThe value 92 results if the pair (1,1) is mistakenly assigned the 100 benefit instead of 50. That one grid cell adds 50/25=2 to the correct mean.
DTreating every positive maximum as a full 100 payment gives an expectation of 96, whose nearest listed value is 95. This discards the intermediate 50 benefit tier.
EIgnoring the cap gives 50 E[max(X,Y)]. The maximum has mean 2.8 on this grid, producing 140 rather than the capped expectation.

Original practice · fully worked

Original variant: award from the smaller rating and a mismatch fee

Two independent reviewers each assign an integer rating chosen uniformly from 0, 1, 2, and 3. An award equals 40 times the smaller rating, plus 20 additional points when the two ratings differ. Calculate the expected award.

A
15 points
B
35 points
C
40 points
D
50 points
E
85 points

Answer and calculation

The expected smaller rating is 14/16=0.875, and the ratings differ with probability 12/16. The expected award is 40(0.875)+20(0.75)=50 points, choice D.

Setup

Let U and V be the two ratings and write the award as the sum of a minimum term and a mismatch term.

A=40\min(U,V)+20\mathbf{1}_{\{U\ne V\}}

Model

Evaluate the expected minimum by its positive integer tail probabilities and count unequal ordered pairs.

\operatorname{E}[\min(U,V)]=\sum_{k=1}^{3}\Pr(U\ge k,V\ge k)

\operatorname{E}[\min(U,V)]=\frac9{16}+\frac4{16}+\frac1{16}=\frac{14}{16}

\Pr(U\ne V)=\frac{12}{16}

Compute

Apply linearity of expectation to the two award components.

\operatorname{E}[A]=40\left(\frac{14}{16}\right)+20\left(\frac{12}{16}\right)=35+15=50

Answer

The mean award is 50 points.

\boxed{50\quad\text{points (D)}}

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How to solve this Joint Distributions question

Setup

Model

Compute

Answer

Probability Multivariate & Risk Models Practice Workbook

Why the other letters tempt

Original variant: award from the smaller rating and a mismatch fee

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P Multivariate Random Variables