## Comprehensive Exam

**Probability and Statistics **

**General information** The exam is to evaluate students' fundamental knowledge of probability and statistics - the bases of their degree. The topics for the exam are a union of the major topics from the Probability (Math 661) and Mathematical Statistics (Stat 754) courses.

**Study guidelines** To study for the exam, we recommend taking both Probability and the Mathematical Statistics classes; practicing by doing problems assigned as homeworks in both courses and more problems from the course text books; doing relevant problems from the actuarial exams; studying proofs of theorems in the texts. Students are expected to know all definitions and theorems with proofs.

We stress that the exam is not based on any particular book. It is an exam based on knowledge of fundamental topics in probability and mathematical statistics. Some texts you may find helpful include:

**Main reference**

DeGroot, M.H. and Schervish, M.J. "*Probability and Statistics*", Addison Wesley, 3rd edition, 2002.

Additional references

Bean, M.A. "*Probability: The Science of Uncertainty with Applications to Investments, Insurance and Engineering*", American Mathematical Society, 2009.

Society of Actuaries (SoA)

**Exam Syllabus**

- The formal language of probability: Random experiment, set theory, sample space, counting and combinatorial methods, probability of union of events, conditional probability, multiplication rule, independent events, the law of total probability and Bayes' theorem.
- Univariate and multivariate random variables and probability

distributions: Discrete, continuous, and mixed distributions; cumulative distribution function; probability density function; probability mass function; quantile function and percentile; marginal and conditional distributions; independence; functions of random variables and random vectors; linear transformations; sums, products, and quotients of random variables; minima and maxima of random variables; order statistics; mixtures and compound distributions and their applications; probability integral transform theorem and random variate generation; Monte-Carlo methods. - Measures of expectation, variation and risk, expected value, geometric mean, median, mean squared and mean absolute error, variance and standard deviation, moments and moment generating function, survival and hazard functions, covariance and correlation, conditional expectation and variance.
- Special discrete and continuous distributions and their

applications: Bernoulli, binomial, Poisson, hypergeometric, multinomial, negative binomial, geometric, exponential, gamma, Weibull, beta, uniform, Pareto, univariate and multivariate normal, lognormal distributions. - Convergence of probability distributions: Convergence in distribution, convergence in probability, and almost sure convergence; Markov and Chebyshev inequalities; the law of large numbers and the central limit theorem; normal approximation to binomial; delta method.
- Sampling distributions related to the normal distribution: The sample mean and its properties; chi-square, student-t, and F distributions; joint distribution of the sample mean and variance.
- Estimation: The method of moments; maximum likelihood estimation and its properties; efficiency, consistency, sufficiency, and unbiasedness; small and large sample confidence intervals; information inequality; loss and risk functions; uniformly minimum variance unbiased (UMVU) estimation; Bayesian estimation.
- Testing hypotheses: Mathematical setup and terminology; power and sample size calculations; p-values; Neyman-Pearson lemma; uniformly most powerful (UMP) tests; likelihood ratio tests, one and two sample z-test and t-test; F-test; Kolmogorov-Smirnov test; chi-square tests of goodness-of-fit; contingency tables and tests for homogeneity.
- Linear models: The method of least squares, linear regression, statistical inference under linear regression model.