Worked solution · Exam P
This Exam P sample reference tests Covariance. This is a variance-of-a-sum calculation with nonzero covariance. The two marginal variances contribute 5 and 8, while covariance enters twice, so the total variance is 5+8+2(3)=19 and choice E.
Independent solution
Setup
Let F and T denote the two loss counts and let S be their combined count.
Model
A sum contains both cross terms from its squared centered expansion.
Compute
Insert the two marginal variances and the covariance.
Answer
The total-count variance is 19.
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
Twenty components are inspected independently. Each is classified as cosmetic rework with probability 0.30, functional rework with probability 0.20, or acceptable otherwise. Let X and Y be the counts in the two rework classes. Calculate Var(X+Y).
For each component, membership in either rework class has probability 0.50. Therefore X+Y is Binomial(20,0.50) and has variance 20(0.50)(0.50)=5, so choice C.
Setup
Combine the two mutually exclusive rework labels into one indicator for whether a component requires any rework.
Model
The combined count is binomial because the twenty component classifications are independent.
Compute
Apply the binomial variance formula.
Answer
The combined rework-count variance is 5.00.
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