Verified solution · Exam P

SOA Exam P Sample Question #574 Solution

This Exam P sample reference tests Joint Distributions. Only the row totals are needed for the strawberry marginal. The readable rows have masses 0.25, 0.33, and 0.24, so normalization fixes the final row at 0.18. These masses give E[S]=2.35 and E[S²]=6.61, hence variance 1.0875 and choice D.

Independent analysis by ActuaryProof · reviewed August 4, 2026

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Independent solution

How to solve this Joint Distributions question

Setup

Sum each readable row of the joint table to obtain the first three marginal probabilities.

\Pr(S=1)=0.25,\qquad \Pr(S=2)=0.33,\qquad \Pr(S=3)=0.24

Model

The unread entries all lie in the same final row, so their individual allocation is irrelevant. Total probability determines that row's combined mass.

\Pr(S=4)=1-(0.25+0.33+0.24)=0.18

Compute

Calculate the first two moments of the resulting marginal distribution.

E[S]=1(0.25)+2(0.33)+3(0.24)+4(0.18)=2.35

E[S^2]=1(0.25)+4(0.33)+9(0.24)+16(0.18)=6.61

\operatorname{Var}(S)=6.61-2.35^2=1.0875

Answer

The variance rounds to 1.09.

\boxed{1.09\quad\text{(D)}}

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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.

AUnderstating the final-row contribution to E[S²] by 0.06 gives 6.55-2.35²=1.0275, which rounds to 1.03.
BUsing the incorrect mean 2.36 while keeping E[S²]=6.61 gives 6.61-2.36²=1.0404.
CUnderstating E[S²] by 0.02 gives 6.59-2.35²=1.0675, which rounds to 1.07.
EThis assumes the allocation among the unread cells is needed. For a function of S alone, only their combined row mass matters, and normalization supplies it.

Original practice · fully worked

Original variant: recovering a missing joint cell

Two discrete variables U and V take values U∈{0,1} and V∈{0,2}. Their joint probabilities are P(U=0,V=0)=0.20, P(U=0,V=2)=0.30, P(U=1,V=0)=x, and P(U=1,V=2)=0.50-x. Given Cov(U,V)=0.10, determine x.

A
0.000
B
0.050
C
0.100
D
0.200
E
0.450

Variant answer in brief

The table gives E[U]=0.5, E[V]=1.6-2x, and E[UV]=1-2x. Therefore Cov(U,V)=0.2-x; equating this to 0.10 gives x=0.10 and choice C.

Setup

Compute the marginal mean of U and express the other required moments in terms of the unknown cell.

E[U]=0.50

E[V]=2(0.30+0.50-x)=1.6-2x

E[UV]=2(0.50-x)=1-2x

Model

Use the covariance identity with the moment expressions from the table.

\operatorname{Cov}(U,V)=E[UV]-E[U]E[V]

Compute

Simplify the covariance and solve the supplied covariance equation.

\operatorname{Cov}(U,V)=1-2x-0.5(1.6-2x)=0.2-x

0.2-x=0.10

x=0.10

Answer

The missing joint probability is 0.10.

\boxed{x=0.10\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #574 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.

How to solve this Joint Distributions question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: recovering a missing joint cell

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution