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
E is correct.

For a Poisson distribution, E[N2]=Var(N)+(E[N])2E[N^{2}] = \text{Var}(N) + (E[N])^{2}. Since Var(N)=3\text{Var}(N) = 3 and E[N]=3E[N] = 3:

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