Syllabus

Comprehensive Examination

Parts –I-A and I-B

Students may review the textbooks that were used in the Probability and Mathematical Statistics sequence (MAT-451, 452 and 453). 

The exams are given in two parts and must be taken on the same day, on a Saturday 9:00 A.M. - 4:00 P.M., in October and/or April/May.

Students who intend to take the exams in a given quarter should notify the program director in writing, early in that quarter.

Part-IA

This is a three-hour closed-book exam.  It covers analytical and theoretical concepts of probability and statistical inference [items 1-5, and major theorems in items 6 and 7 of the following list of topics]. Normally, this exam is given between 9:00 A.M.-12:00 noon.

Part-IB

This is a three-hour open-book exam.  It covers topics of statistical inference described in items 6-11 of the following list of topics.  Notes, books, tables, and calculators may be used.  The exam is given between 1:00-4:00 P.M. the same day as part I-A.

List of Major Topics

1.                     Elements of probability.

2.                     Probability distributions: their moments, moment-generating functions, functions of random variables, and transformation of variables.

3.                     Discrete Probability models: Bernoulli, Binomial, Poisson, Geometric, negative binomial, hyper-geometric, and multinomial distributions.

4.                     Continuous Probability Models: Uniform, Normal, the Gamma family of distributions [including the exponential & χ²];  the Beta-family, Student-t, and Fisher’s F-distributions.

5.                     Sampling distributions of means, variances, proportions.  Laws of large numbers, Chebyshev’s Theorem, and the Central Limit Theorem.

6.                     Estimation:  Properties of estimators, methods of estimation, point and interval estimation.

7.                     Hypothesis testing: Neyman-Pearson lemma, most powerful tests, simple and composite hypothesis tests of means, variances and proportions.

8.                     The Analysis of Variance (one-way ANOVA).

9.                     χ² -tests of Goodness of Fit, tests of Independence and Homogeneity.

10.                 Nonparametric methods of inference: One and two-sample Wilcoxon-Mann-Whitney and Sign-tests; Kruskell-Wallis K-sample test;  Runs test, and Spearman’s Correlation.