Exam P technique guide
Learn how to recognize the method, recall the formulas that control it, and work through 62 sample-question solutions.
Try this original problem before opening the solution. Then compare the method with SOA Exam P sample question 17.
Original practice · fully worked
Independent measurements U and V are normal with means 2 and 5 and variances 1 and 4. A composite score is T=2U+V. Determine the 90th percentile of T.
The composite is normal with mean 9 and variance 8. Adding 1.28155 standard deviations gives the 90th percentile 12.625, choice D.
Setup
Compute the mean of T=2U+V by linearity of expectation.
Model
Independence makes the variance of T the sum of the scaled component variances, so T is normal with mean 9 and variance 8.
Compute
Add the 90th-percentile standard-normal quantile 1.28155 times the standard deviation √8 to the mean, obtaining 12.624775.
Answer
The 90th percentile of the composite score rounds to 12.625, corresponding to choice D.
Technique orientation
Normal-distribution work is mostly translation: convert the requested event to a standard-normal event, preserve the direction of each inequality, and use symmetry or a quantile only after the standardization is visible. A sketch of the relevant tail is often enough to catch a sign error before table lookup.
Exam P uses the normal model directly, through linear combinations, and as an approximation to counts or aggregate quantities. The solution should distinguish variance from standard deviation, identify whether continuity correction is part of the stated approximation, and avoid rounding the standardized value too early.
Recognition audit
Before calculation
Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.
Formula core
These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.
Standardization
σ is the standard deviation, not the variance.
Interval probability
Continuous endpoints do not change the probability, but both tails must use the same scale.
Linear combination
Delete the covariance term only after independence or zero covariance is justified.
Three failure modes to catch
Verified practice ledger
The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.
1 verified reference
39 verified references
22 verified references
Continue the study loop
The complete Exam P manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.
Related drills
Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.
103 verified problems
Conditioning, total probability, and Bayes calculations organized around the denominator that defines the updated sample space.
83 verified problems
Waiting-time probabilities, parameter conversions, memorylessness, moments, and insurance-payment applications.
72 verified problems
Count probabilities, rate scaling, aggregation, thinning, and links between Poisson counts and waiting times.
57 verified problems
Second moments, transformations, sums, covariance terms, and conditional variance decompositions.
55 verified problems
Event independence, factorization, independent trials, and the consequences independence does—and does not—justify.
50 verified problems
Continuous and discrete uniform models, interval geometry, transformations, order statistics, and moments.
46 verified problems
Tail-integral proof, survival-function expectation, discrete tail sums, loss payments, linearity, and conditioning.
