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

Strengthen your command of expectation and variance for Exam P. These questions cover moments, moment generating functions, conditional expectation, and variance decomposition formulas.

118 Questions
70 Easy
29 Medium
19 Hard
2026 Syllabus
100% Free

Sample Questions

Question 1 Easy
N ~ Poisson(3). Find E[N²].
Solution
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
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
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 118 Questions →

More Exam P Topics

Probability Fundamentals Conditional Probability & Bayes Discrete Distributions Continuous Distributions Joint & Marginal Distributions Insurance Applications

About FreeFellow

FreeFellow is a free exam prep platform for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, and securities licensing candidates. Every question includes a detailed solution. Full lessons, flashcards with spaced repetition, timed mock exams, performance analytics, and a personalized study plan are all included — no paywalls, no ads.