Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Parametric Estimation Practice Questions

Parametric estimation on SOA Exam FAM covers maximum likelihood estimation (MLE), method of moments, and goodness-of-fit testing for insurance loss distributions. These techniques are foundational to actuarial modeling.

48 questions 22 easy 21 medium 5 hard 2026 syllabus

Sample Questions

Question 1 Easy
Five observations from a Uniform(0, θ\theta) distribution are: 2.1, 3.5, 1.8, 4.2, 2.9. What is the MLE of θ\theta?
Solution
A is correct.

For the Uniform(0, θ\theta) distribution, the MLE of θ\theta is the maximum order statistic: θ^=X(n)=max⁡(2.1,3.5,1.8,4.2,2.9)=4.2\hat{\theta} = X_{(n)} = \max(2.1, 3.5, 1.8, 4.2, 2.9) = 4.2.
Question 2 Medium
A complete sample of 25 exponential losses has total 10,000. The model has unknown mean θ\theta. Using maximum likelihood, estimate the variance of a loss from this model.
Solution
D is correct.

The exponential log-likelihood is −25ln⁡θ−10,000/θ-25\ln\theta-10{,}000/\theta, maximized at θ^=10,000/25=400\hat\theta=10{,}000/25=400. The model variance is θ2\theta^2. By invariance, its maximum likelihood estimate is θ^2=160,000\hat\theta^2=160{,}000. This is the variance of a loss, not the variance of an estimator.
Question 3 Hard
Loss amounts follow a Pareto distribution with pdf f(x)=αθα(x+θ)α+1f(x) = \frac{\alpha\theta^\alpha}{(x+\theta)^{\alpha+1}} for x>0x > 0. Both α\alpha and θ\theta are unknown. A sample of 3 observations is: 100, 400, 500. Write the system of equations that the MLEs must satisfy.
Solution
E is correct.

The log-likelihood is: ℓ(α,θ)=nln⁡α+nαln⁡θ−(α+1)∑ln⁡(xi+θ)\ell(\alpha, \theta) = n\ln\alpha + n\alpha\ln\theta - (\alpha+1)\sum\ln(x_i + \theta). Taking partial derivatives: ∂ℓ∂α=nα+nln⁡θ−∑ln⁡(xi+θ)=0\frac{\partial\ell}{\partial\alpha} = \frac{n}{\alpha} + n\ln\theta - \sum\ln(x_i + \theta) = 0. ∂ℓ∂θ=nαθ−(α+1)∑1xi+θ=0\frac{\partial\ell}{\partial\theta} = \frac{n\alpha}{\theta} - (\alpha+1)\sum\frac{1}{x_i + \theta} = 0. With n=3n = 3 and the given data: Equation 1: 3α+3ln⁡θ−[ln⁡(100+θ)+ln⁡(400+θ)+ln⁡(500+θ)]=0\frac{3}{\alpha} + 3\ln\theta - [\ln(100+\theta) + \ln(400+\theta) + \ln(500+\theta)] = 0. Equation 2: 3αθ−(α+1)[1100+θ+1400+θ+1500+θ]=0\frac{3\alpha}{\theta} - (\alpha+1)\left[\frac{1}{100+\theta} + \frac{1}{400+\theta} + \frac{1}{500+\theta}\right] = 0. This system must be solved simultaneously using numerical methods (e.g., Newton-Raphson). From Equation 1, we can express α\alpha as a function of θ\theta and substitute into Equation 2. From Equation 1: α=3∑ln⁡(xi+θ)−3ln⁡θ\alpha = \frac{3}{\sum\ln(x_i+\theta) - 3\ln\theta} (profile likelihood approach).
Create a Free Account to Access All 48 Questions →

Browse Exam FAM practice samples, free with step-by-step solutions and no signup.

Guides & Articles

Free Exam FAM Sample Questions: 262 SOA Items Across Short- and Long-Term (2026) Which Actuarial Exam Should You Take First? (2026 Order Guide) SOA Exam FAM Explained: 34 Questions, 57% Pass Rate, Short vs Long-Term (2026) How to Pass Exam FAM in 4 Months: Study Plan + Pitfalls (2026)

About FreeFellow

Jeffrey Ting, founder of FreeFellow
Jeffrey Ting
FSA, CFA · Founder

FreeFellow was built by Jeffrey Ting, a credentialed actuary and CFA charterholder who has passed thirteen of the hardest exams in finance, all on his first attempt. He paid four-figure prep fees along the way.

So he started writing his own questions, then lessons, then mock exams, until it grew into a full prep platform covering 40 exams with more than 45,000 original practice questions.

01
Cost shouldn't decide who gets in.

A CFA charter, a CPA license, an actuarial credential. Each opens a real career, but prep fees add to the cost of getting there. FreeFellow keeps its original question bank and written lessons free so you can study even if a paid course is out of reach.

02
Free should mean free.

The original question bank, the worked solutions, the topic lessons, mixed practice, and your readiness score stay free, forever. Full access needs a free account, and practice usage limits apply. Fellow adds AI grading on supported exams, spaced-repetition flashcards, full-length mock exams, analytics, and a study plan that adapts to your progress.

Meet the founder ›Read the values ›