Exam P practice problems: Order statistics, sums and the Central Limit Theorem

10 problems · Untimed · Worked solution after each answer

The minimum and maximum of independent variables, probabilities and moments of linear combinations, and normal approximations by the Central Limit Theorem. This drill covers topic 3 of the Exam P syllabus, Multivariate Random Variables (23–30% of the exam).

3f Order statistics
3g Probabilities for linear combinations
3h Moments of linear combinations
3i Central Limit Theorem

Line drawing of a row of five light bulbs, one of them dark, beside a stopwatch

Before you start

Every problem has five options, A to E, and one correct answer, as on the real exam. Every value you need is in the problem, so you can work it with an approved calculator; the worked solution shows each step and names the mistake behind the most tempting wrong answer.
Check each answer to see the explanation before moving on. Checked answers cannot be changed.

Study guide: Order statistics, sums and the Central Limit Theorem

For independent variables, P(max ≤ x) = ∏ F_i(x) and P(min > x) = ∏ S_i(x); the minimum of independent exponentials is exponential with the rates added.
Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y); the covariance term drops out only for independent variables.
The sum of n independent copies has mean nμ and variance nσ²; the average has variance σ²/n.
For a CLT approximation, standardize the sum, not a single variable, and use a continuity correction only when the stem asks for it.

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Original practice problems written to the published SOA syllabus; not SOA sample questions. Independent preparation, not affiliated with or endorsed by the Society of Actuaries (SOA).