Worked solution · Exam P
This Exam P sample reference tests Linear Combinations of Independent Random Variables. Each observation has variance 9. Independence makes the variance of the linear combination equal to 3²(9)+9+9+9=108, so choice E is correct.
Independent solution
Setup
Convert the common standard deviation into variance.
Model
For independent observations, square every coefficient and add the resulting variance contributions.
Compute
Substitute the common variance.
Answer
The variance is 108.
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
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
Independent calibration errors A and B have variances 4 and 9. A combined error is Q=kA-B, where k is positive. The standard deviation of Q is 5. Calculate k.
The target standard deviation gives variance 25. Independence yields Var(Q)=4k²+9, so 4k²+9=25 and the positive coefficient is k=2, which is choice B.
Setup
Convert the given standard deviation of the combined error to variance.
Model
Use independence and square the linear-combination coefficients.
Compute
Solve the variance equation and retain the specified positive coefficient.
Answer
The positive calibration coefficient is 2.
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 Variance problems
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