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

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.
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More Exam P Topics

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

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