Free SOA Exam P (Probability) Expectation & Variance Practice Questions

Expectation and variance on SOA Exam P cover moments, moment generating functions, conditional expectation, and the variance decomposition formula Var(X) = E[Var(X|Y)] + Var(E[X|Y]). These appear in many applied problems.

113 Questions
68 Easy
29 Medium
16 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
N ~ Poisson(3). Find E[N²].
Solution
B is correct.

E[N2]E[N^2] = Var(N)\text{Var}(N) + (E[N]E[N])^2 = 3 + 9 = 12.
Question 2 Medium
X ~ Bernoulli(p). Find E[X³].
Solution
B is correct.

Since X only takes values 0 and 1: X^3 = X. Therefore E[X3]E[X^3] = E[X]E[X] = p
Question 3 Hard
N ~ Poisson(3). Given N=n, X|N ~ Binomial(n, 0.4). Calculate E[X] and Var(X).
Solution
D is correct.

E[X]=E[0.4N]=1.2E[X] = E[0.4N] = 1.2. E[Var(X∣N)]=E[0.24N]=0.72E[\text{Var}(X|N)] = E[0.24N] = 0.72. Var(E[X∣N])=0.16⋅Var(N)=0.48\text{Var}(E[X|N]) = 0.16 \cdot \text{Var}(N) = 0.48. Var(X)=0.72+0.48=1.20\text{Var}(X) = 0.72 + 0.48 = 1.20.
Create a Free Account to Access All 113 Questions →

About FreeFellow

FreeFellow is an AI-native exam prep platform for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, GARP FRM, IRS Enrolled Agent, IMA CMA, and FINRA / NASAA securities licensing candidates — built around modern AI as a core capability rather than as a bolt-on. Every lesson ships with AI-narrated audio. Every constructed-response item has a copy-to-AI prompt builder so candidates can paste their answer into their own ChatGPT or Claude for self-graded feedback. Fellow members get instant AI grading on essays against the official rubric (currently CFA Level III, expanding to other essay-bearing sections).

The 70% you need to pass — question bank, written solutions, lessons, formula sheet, mixed practice, readiness tracking — is free forever, with no trial period and no credit card. Become a Fellow ($59/quarter or $149/year per track) to unlock mock exams, flashcards with spaced repetition, performance analytics, AI essay grading, and a personalized study plan.