Worked solution · Exam P
This Exam P sample reference tests Conditional Variance. For S uniform from 5 to 20, E[S]=12.5, Var(S)=18.75, and E[S²]=175. Total variance gives E[9S²]+Var(3S)=1,575+168.75=1,743.75, choice E.
Independent solution
Setup
Record the conditional moments of the exponential variable in terms of its random scale S.
Model
Compute the moments needed from the continuous uniform distribution.
Compute
Use the within-scale and between-scale pieces of the law of total variance.
Answer
The unconditional variance is 1,743.75.
Test the method on unseen problems
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.
Error diagnosis
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Original practice · fully worked
A production regime R takes value 1 with probability 0.25, value 3 with probability 0.35, and value 6 otherwise. Conditional on R, the defect count Y is Poisson with mean 2R. Calculate Cov(R,Y).
The regime has E[R]=3.7, E[R²]=17.8, and Var(R)=4.11. Because E[Y|R]=2R, total covariance gives Cov(R,Y)=Cov(R,2R)=2 Var(R)=8.22, choice C.
Setup
Compute the first two moments of the discrete regime variable.
Model
Conditioning makes R fixed, so the conditional covariance with Y is zero; only the changing conditional mean contributes.
Compute
Replace the conditional mean by its linear expression.
Answer
The regime and defect count have covariance 8.22.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 140 Exam P multivariate random variables practice problems.
Drill the method: Exam P Uniform Distribution problems
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