Exam P practice problems: Discrete distributions

10 problems · Untimed · Worked solution after each answer

Binomial, geometric, negative binomial, hypergeometric, Poisson and discrete uniform variables: probabilities, means, variances and percentiles. This drill covers topic 2 of the Exam P syllabus, Univariate Random Variables (44–50% of the exam).

2a Random variables, probability functions and CDFs
2b Conditional probabilities
2c Expected values, moments, mode, median and percentiles
2d Variance, standard deviation and coefficient of variation

Line drawing of three coins tossed above an open hand beside a handheld tally counter

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: Discrete distributions

Match the story to the family: a fixed number of trials is binomial, trials until the first success geometric, until the r-th success negative binomial, sampling without replacement hypergeometric, counts per period Poisson.
Check which geometric convention the stem uses: number of failures before the first success, or number of trials including it. The means differ by 1.
For a Poisson count, P(N ≥ 2) = 1 − P(0) − P(1); E[N] = Var(N) = λ, and λ scales with the length of the period.
Var(X) = E[X²] − (E[X])²; the coefficient of variation is σ/μ.

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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).