Free SOA Exam P (Probability) Insurance Applications Practice Questions

Insurance and risk management applications on SOA Exam P connect probability theory to actuarial practice. Questions involve deductibles, policy limits, loss distributions, and expected claim payments using loss elimination ratios.

125 Questions
62 Easy
38 Medium
25 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
Losses are lognormal with ln⁡(X)∼N(7,4)\ln(X) \sim N(7, 4). Calculate E[X]E[X].
Solution
B is correct.

E[X]=eμ+σ2/2=e7+2=e9E[X] = e^{\mu + \sigma^2/2} = e^{7 + 2} = e^{9}.
Question 2 Medium
N ~ Poisson(10), each claim Cᵢ iid Exp(mean 500), independent of N. S = ΣCᵢ. Calculate Var(S).
Solution
B is correct.

Compound Poisson: Var(S)=λ⋅E[C2]\text{Var}(S) = \lambda \cdot E[C^2]. E[C2]=Var(C)+(E[C])2=250,000+250,000=500,000E[C^2] = \text{Var}(C) + (E[C])^{2} = 250{,}000 + 250{,}000 = 500{,}000. Var(S)=10(500,000)=5,000,000\text{Var}(S) = 10(500{,}000) = 5{,}000{,}000
Question 3 Hard
X ~ Uniform(0, 500). Policy: deductible 100, max payment 300. Calculate Var(payment).
Solution
A is correct.

Y = min(max(X−100,0), 300). Cap at X=400. E[Y]E[Y] = ((1)500\frac{(1)}{500})(121^{2}/2) + (300)5\frac{(300)}{5} = 90 + 60 = 150. E[Y2]E[Y^2] = ((1)500\frac{(1)}{500})(131^{3}/3) + 121^{2}/5 = 18000 + 18000 = 36000. Var = 36000 − 22500 = 13500.
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