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 questions68 easy30 medium15 hard2026 syllabus
Sample Questions
Question 1
Easy
N ~ Poisson(3). Find E[N²].
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Correct Answer: E
Solution
E is correct.
For a Poisson distribution, E[N2]=Var(N)+(E[N])2. Since Var(N)=3 and E[N]=3:
E[N2]=Var(N)+(E[N])2=3+9=12.
Question 2
Medium
X ~ Bernoulli(p). Find E[X³].
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Correct Answer: E
Solution
E is correct.
Since X only takes values 0 and 1: X^3 = X. Therefore E[X3] = E[X] = p
Question 3
Hard
N ~ Poisson(3). Given N=n, X|N ~ Binomial(n, 0.4). Calculate E[X] and Var(X).
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