Exam P technique guide
Learn how to recognize the method, recall the formulas that control it, and work through 57 sample-question solutions.
Try this original problem before opening the solution. Then compare the method with SOA Exam P sample question 55.
Original practice · fully worked
A chemical reading X has variance 16 in laboratory units. The published reading is Y=2.5X+7. What is the variance of Y?
The additive calibration does not affect variance, while multiplication by 2.5 contributes a factor of 6.25, giving 100.
Setup
The published reading is the affine transformation Y=2.5X+7 of a reading whose variance is 16.
Model
The additive calibration 7 does not affect dispersion, while the multiplicative coefficient 2.5 enters the variance as its square.
Compute
Since 2.5 squared is 6.25, the transformed variance is 6.25 × 16, or 100.
Answer
Thus Var(Y)=100 squared laboratory units, corresponding to choice A.
Technique orientation
Variance measures squared dispersion around the mean, so it is not linear in the way expectation is. Most errors disappear if the mean and second moment are computed separately, or if a linear-combination formula is written with every covariance term before any independence assumption is used.
Exam P variance questions range from direct distribution moments to insurance payments, mixtures, and aggregates. A clean solution identifies the random variable whose variance is requested, applies transformations to that variable rather than to its standard deviation, and uses conditional decomposition when a latent group changes the mean.
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.
Computational identity
Square the mean after taking expectation; it is not E[X²].
Linear combinations
The coefficient on each variance is squared.
Law of total variance
The second term captures variation in conditional means across groups.
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
32 verified references
24 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.
62 verified problems
Standardization, inverse percentiles, linear combinations, normal approximations, and tail control.
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.
