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) \sim N(7, 4)

. Calculate

E[X]

Solution

D is correct.

E[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:

\text{Var}(S)

\lambda \cdot E[C^2]

E[C^2]

\text{Var}(C)

+ (

E[C]

)^2 = 250,000 + 250,000 = 500,000.

\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

C is correct.

Y = min(max(X−100,0), 300). Cap at X=400.

E[Y]

= (

\frac{(1)}{500}

)(

1^{2}

/2) +

\frac{(300)}{5}

= 90 + 60 = 150.

E[Y^2]

= (

\frac{(1)}{500}

)(

1^{3}

/3) +

1^{2}

/5 = 18000 + 18000 = 36000. Var = 36000 − 22500 = 13500.

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