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Which density form correctly defines a member of the exponential dispersion family used in GLMs?

f(y;θ,φ) = exp((yθ - b(θ))/a(φ) + c(y,φ))

f(y;θ,φ) = (1/√(2πφ)) exp(-(y-θ)²/(2φ))

f(y;θ,φ) = θ^y (1-θ)^(1-y) for y in {0,1}

f(y;θ,φ) = exp(-θy) for y > 0

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2026 Statistics

Key Facts: MAS-II Exam

~45

Exam Questions

CAS MAS-II content outline

4 hours

CBT Exam Time

Pearson VUE

~$1,000

Exam Fee

CAS fee schedule

~30%

Historical Pass Rate

CAS-published results

25%

Largest Domain (GLMs)

MAS-II syllabus

2x/year

2026 Sittings

Apr 22-May 1, Oct 28-Nov 5

MAS-II is a 4-hour CBT exam with about 45 questions and a historical pass rate near 30%. The 2026 sittings run April 22 - May 1 (Spring) and October 28 - November 5 (Fall). The exam fee is approximately $1,000. Generalized Linear Models (25%) and credibility theory — Bühlmann (15%) plus Bühlmann-Straub and empirical Bayes (10%) — make up half the syllabus, with Bayesian analysis (15%), tree-based models (15%), clustering (10%), PCA (5%), and cross-validation (5%) filling the rest.

Sample MAS-II Practice Questions

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1Which density form correctly defines a member of the exponential dispersion family used in GLMs?

A.f(y;θ,φ) = exp((yθ - b(θ))/a(φ) + c(y,φ))

B.f(y;θ,φ) = (1/√(2πφ)) exp(-(y-θ)²/(2φ))

C.f(y;θ,φ) = θ^y (1-θ)^(1-y) for y in {0,1}

D.f(y;θ,φ) = exp(-θy) for y > 0

Explanation: The exponential dispersion family used in GLMs has density exp((yθ - b(θ))/a(φ) + c(y,φ)), where θ is the canonical parameter, b(θ) is the cumulant function, and φ is the dispersion parameter. The other choices are specific distributions, not the general family form.

2For a Poisson GLM, what is the canonical link function?

A.log(μ)

B.μ

C.logit(μ)

D.1/μ

Explanation: The canonical link for the Poisson distribution is the log link, log(μ). The identity link goes with Normal, the logit link with Binomial, and the inverse link 1/μ with the Gamma distribution.

3Which link function is canonical for a Binomial GLM (logistic regression)?

A.Logit: log(μ/(1-μ))

B.Log: log(μ)

C.Identity: μ

D.Inverse: 1/μ

Explanation: The canonical link for the Binomial distribution is the logit, log(μ/(1-μ)). It maps probabilities in (0,1) to the entire real line so the linear predictor Xβ is unconstrained.

4Which link function is canonical for a Gamma GLM?

A.Inverse: 1/μ

B.Log: log(μ)

C.Identity: μ

D.Logit: log(μ/(1-μ))

Explanation: The canonical link for the Gamma distribution is the inverse link, 1/μ. In actuarial practice the log link is often used instead because it guarantees positive predictions and gives multiplicative interpretations of coefficients.

5An actuary fits a Poisson GLM for claim counts and wants to adjust for policy exposure (years on risk). What is the standard treatment?

A.Add log(exposure) as an offset with coefficient fixed at 1

B.Add exposure as an additional covariate

C.Multiply the response by exposure before fitting

D.Use exposure as the dispersion parameter φ

Explanation: For a Poisson model with log link, exposure is included as an offset: log(exposure) is added to the linear predictor with coefficient fixed at 1. This models claim frequency per unit of exposure while keeping the response on its original count scale.

6Which distribution from the exponential family is most commonly used to model pure premium (loss per exposure) in actuarial GLMs?

A.Tweedie

B.Normal

C.Binomial

D.Inverse Gaussian

Explanation: The Tweedie distribution with power parameter p between 1 and 2 is a compound Poisson-Gamma model. It naturally has a point mass at zero for policies with no claims and a continuous Gamma-like distribution for positive losses, making it ideal for pure premium modeling.

7A Poisson GLM shows estimated dispersion φ̂ = 2.5. What is the appropriate response?

A.Refit using a quasi-Poisson model to inflate standard errors

B.Drop the offset from the model

C.Switch to an identity link

D.Reduce the number of predictors until φ̂ = 1

Explanation: Estimated dispersion well above 1 indicates overdispersion relative to Poisson. A quasi-Poisson refit keeps the Poisson mean structure but inflates standard errors by √φ̂ to give honest inference. Dropping predictors does not address the underlying variance structure.

8Which expression defines the deviance D used to compare nested GLMs?

A.D = 2(LL_saturated - LL_model)

B.D = LL_model - LL_null

C.D = -2 LL_model

D.D = LL_saturated / LL_model

Explanation: The deviance is twice the difference between the log-likelihood of the saturated model (one parameter per observation) and the fitted model. For nested models, the difference in deviances is asymptotically chi-square with degrees of freedom equal to the number of constrained parameters.

9Two nested Poisson GLMs differ by 3 parameters. The deviance drops from 250 to 240. At the 5% level, the chi-square critical value with 3 df is 7.81. What is the conclusion?

A.The larger model fits significantly better; reject the smaller model

B.The smaller model fits significantly better; reject the larger model

C.The models are equivalent; choose based on AIC alone

D.The test is invalid because deviance must always increase

Explanation: The likelihood-ratio test statistic equals the change in deviance, 250 - 240 = 10, which exceeds the chi-square critical value of 7.81 with 3 df. We therefore reject the smaller model in favor of the larger one at the 5% level.

10Which formula correctly defines the Pearson chi-square statistic for a GLM?

A.Σ (y_i - μ̂_i)² / V(μ̂_i)

B.Σ (y_i - μ̂_i) / μ̂_i

C.Σ y_i log(y_i / μ̂_i)

D.Σ (y_i - ȳ)²

Explanation: The Pearson chi-square statistic standardizes squared residuals by the GLM variance function V(μ̂_i). For a Poisson GLM V(μ) = μ, so each term reduces to (y_i - μ̂_i)² / μ̂_i. Dividing the statistic by residual degrees of freedom is a common dispersion estimate.

About the MAS-II Exam

MAS-II is the second CAS modern statistics exam on the ACAS pathway. It tests generalized linear models, Bayesian analysis, Bühlmann and Bühlmann-Straub credibility, empirical Bayes, tree-based statistical learning, cluster analysis, principal components analysis, and cross-validation. Candidates must combine actuarial credibility intuition with modern statistical learning judgment.

Questions

45 scored questions

Time Limit

4 hours (CBT)

Passing Score

Scaled

Exam Fee

~$1,000 (Casualty Actuarial Society (CAS))

MAS-II Exam Content Outline

25%

Generalized Linear Models

Exponential family, canonical link functions, deviance, Pearson chi-square, AIC, BIC, offsets for exposure, Tweedie pure premium models, and quasi-Poisson for overdispersion

15%

Bayesian Analysis

Bayes' theorem, conjugate priors (Beta-Binomial, Gamma-Poisson, Normal-Normal), posterior summaries, credible intervals, and posterior predictive distributions

15%

Bühlmann Credibility

Hypothetical mean μ(θ), process variance v(θ), expected process variance v, variance of hypothetical means a, K = v/a, Z = N/(N+K), and Bayesian credibility connections

10%

Bühlmann-Straub & Empirical Bayes

Unequal exposures, ZBS = nP/(nP+K), partial credibility weighting, and nonparametric empirical Bayes estimation of v and a from data

15%

Tree-Based Models

Decision trees with Gini, entropy, and MSE splits, cost-complexity pruning, random forest mtry and OOB error, AdaBoost, gradient boosting, and XGBoost regularization

10%

Cluster Analysis

K-means within-cluster sum of squares, K-means++ initialization, elbow and silhouette diagnostics, agglomerative hierarchical linkages (single, complete, average, Ward), and dendrogram cuts

Principal Components Analysis

Eigendecomposition of the covariance matrix, scree plots, cumulative variance explained, biplots, and varimax rotation

Cross-Validation & Model Selection

k-fold cross-validation (k=5 or 10), stratified and time-series CV without shuffling, leave-one-out, train/val/test splits, and the bias-variance tradeoff

How to Pass the MAS-II Exam

What You Need to Know

Passing score: Scaled
Exam length: 45 questions
Time limit: 4 hours (CBT)
Exam fee: ~$1,000

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

MAS-II Study Tips from Top Performers

1Master the GLM exponential-family form and the canonical links: identity for Normal, log for Poisson, logit for Binomial, and inverse for Gamma

2Drill the Bühlmann formulas until K = v/a and Z = N/(N+K) feel automatic, and connect them to Bayesian credibility under conjugate priors

3Practice empirical Bayes by estimating v (expected process variance) and a (variance of hypothetical means) directly from sample data

4Build intuition for tree-based models by comparing bagging, random forests, and boosting on the same dataset and tracking OOB error

5Use cross-validation correctly: k=5 or 10 for most problems, stratified for classification, and time-series CV without shuffling for temporal data

Frequently Asked Questions

What is the format of CAS MAS-II?

MAS-II is a 4-hour computer-based test administered by Pearson VUE for the Casualty Actuarial Society. The exam contains about 45 questions, primarily multiple choice but with possible multiple-selection, point-and-click, fill-in-the-blank, and matching items. The 2026 sittings run April 22 to May 1 (Spring) and October 28 to November 5 (Fall).

What topics are tested on MAS-II?

The MAS-II syllabus covers Generalized Linear Models (25%), Bayesian Analysis (15%), Bühlmann Credibility (15%), Bühlmann-Straub and Empirical Bayes (10%), Tree-Based Models (15%), Cluster Analysis (10%), Principal Components Analysis (5%), and Cross-Validation and Model Selection (5%). Together GLMs and credibility theory account for roughly half the exam.

What is the MAS-II pass rate?

Historical MAS-II pass rates have hovered around 30%, making it one of the more challenging early CAS exams. CAS now reports results on a 0-10 scale with 6-10 marking a pass and no longer publishes a fixed numeric pass mark. Candidates should expect a steep curve and plan their study time accordingly.

How much does MAS-II cost?

The MAS-II exam fee is approximately $1,000 for most candidates, paid through the CAS portal. There may be additional study material and prep-provider costs on top of the registration fee. Always confirm the current fee on the CAS website before registering, since CAS adjusts the schedule periodically.

How long should I study for MAS-II?

Most candidates plan 300 to 500 hours of dedicated study for MAS-II, spread across 16 to 26 weeks. The biggest blocks of time should go to Generalized Linear Models and credibility theory, since they make up half the exam. Statistical learning topics (trees, clustering, PCA) and cross-validation round out the syllabus and reward consistent practice.

What prerequisites should I have before MAS-II?

CAS recommends passing MAS-I and having strong statistical maturity before sitting MAS-II. You should be comfortable with probability distributions, maximum likelihood estimation, regression, and basic R or Python output. Without that base, GLM diagnostics, Bayesian updating, and empirical Bayes credibility derivations become very hard to digest under exam pacing.