Verified solution · Exam P
This Exam P sample reference tests Covariance. Recover Var(X), Var(Y), and Cov(X,Y), then expand Cov(X+Y,X+1.2Y). The result is 8.8, official choice A.
Independent solution
Setup
Recover each variance by subtracting the square of the corresponding mean from its second raw moment.
Model
Use the known variance of the original sum to solve for the covariance between the two components.
Compute
Expand the requested covariance bilinearly. Substituting the two variances and recovered covariance gives 8.8.
Answer
The requested covariance is 8.80, selecting choice A.
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Original practice · fully worked
Sensor outputs X and Y have variances 4 and 9 and covariance 2. Define A=X+2Y and B=3X-Y. Find Cov(A,B).
Bilinearity gives 3Var(X)-2Var(Y)+(6-1)Cov(X,Y)=12-18+10=4.
Setup
Write each new sensor score as a linear combination of the original outputs.
Model
Expand covariance bilinearly, retaining the two variance terms and both cross-covariance terms with their signs.
Compute
Substituting the stated variances and covariance gives 4.
Answer
The covariance of the two sensor scores is 4, selecting choice A.
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ActuaryProof. “SOA Exam P Sample Question #80 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
