Verified solution · Exam P

SOA Exam P Sample Question #535 Solution

This Exam P sample reference tests Linear Combinations of Independent Random Variables. The unreimbursed proportions are 0.20 and 0.10. Independence makes the total variance 0.20²(40,000)+0.10²(10,000)=1,700, so choice A is correct.

Independent analysis by ActuaryProof · reviewed August 4, 2026

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Independent solution

How to solve this Linear Combinations of Independent Random Variables question

Setup

Let H and D denote the two losses. Convert each reimbursement rate into the fraction retained by the policyholder.

U=0.20H+0.10D

Model

For independent variables, the variance of a linear combination is the sum of the squared coefficients times the component variances.

\operatorname{Var}(U)=0.20^2\operatorname{Var}(H)+0.10^2\operatorname{Var}(D)

Compute

Insert the two supplied variances.

\operatorname{Var}(U)=0.04(40{,}000)+0.01(10{,}000)

\operatorname{Var}(U)=1{,}600+100=1{,}700

Answer

The variance of total unreimbursed loss is 1,700.

\boxed{1{,}700\quad\text{(A)}}

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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.

BThis weights the two variances by 0.20 and 0.10 instead of by the squared coefficients: 0.20(40,000)+0.10(10,000)=9,000.
CThis subtracts the reimbursement variance from the original total variance, 50,000-33,700=16,300, as though the loss and its reimbursement were independent.
DThis uses reimbursement fractions rather than unreimbursed fractions, giving 0.8²(40,000)+0.9²(10,000)=33,700.
EThis simply adds the original loss variances, 40,000+10,000, ignoring both reimbursement transformations.

Original practice · fully worked

Original variant: variance of a blended sensor estimate

Two independent sensors measure different environmental quantities. Their readings X and Y have variances 225 and 64. A calibration score is defined as S=0.40X-0.75Y. Calculate Var(S).

A
0
B
36
C
72
D
144
E
289

Variant answer in brief

Independence removes the covariance term, so Var(S)=0.40²(225)+0.75²(64)=36+36=72. Choice C is correct.

Setup

Identify the coefficients in the stated linear combination.

S=0.40X-0.75Y

Model

The sign of a coefficient disappears when it is squared in a variance, and independence makes the covariance term zero.

\operatorname{Var}(S)=0.40^2\operatorname{Var}(X)+(-0.75)^2\operatorname{Var}(Y)

Compute

Evaluate the two contributions separately.

0.16(225)=36,\qquad 0.5625(64)=36

\operatorname{Var}(S)=36+36=72

Answer

The calibration score has variance 72.

\boxed{72\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #535 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Linear Combinations of Independent Random Variables question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: variance of a blended sensor estimate

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution