**Ph.D. Qualifying Exams**

A Ph.D. student must pass two qualifying exams, one in Algebra and one in Analysis, before becoming a Ph.D. candidiate. Each part may be attempted at most three times.

**Upcoming Qualifying Exam Schedule for Spring **** 2016:**

**Algebra:** Friday, January 15, 2016

SE 215, 2pm to 5pm

**Probability/Statistics**:
Tuesday, January 19, 2016

SE 215, 2pm to 5pm

**Analysis: ** Tuesday, January 21, 2016

SE 215, 2pm to 5pm

**
Algebra Exam:** group theory, Sylow theorems, the structure of finitely-generated abelian groups, ring theory, Euclidean rings, UFDs, polynomial rings, vector spaces, modules, linear transformations, eigenvalues, minimal polynomials, matrices of linear transformations, Galois theory, and finite fields.

**Analysis Exam: **the real numbers, metric space topology, uniform convergence, Arzela-Ascoli Theorem, differentiation and Riemann integration of single-variable functions, power series, Stone-Weierstrass Theorem, measure theory, Lebesgue integral, convergence theorems for the Lebesgue integral, absolute continuity, the Fundamental Theorem of Calculus.

**Probability & Statistics Exam**: Advanced topics in Probability and Statistics: Borel-Cantelli lemma, normal and Poisson distributions, Chi-square and exponential distributions, t and F distributions, Markov and Chebyshev inequalities, convergence in distribution, in probability and almost surely, law of large numbers, central limit theorem, delta method, Slutsky lemma, LSE, MLE, BLUE, sufficient statistics, Cramer-Rao inequality, Fisher information matrix, hypothesis tests via likelihood ratio test and Bayes test.

The** Probability & Statistics Exam** will be divided into three parts. Total 3 hours.

**Part 1.** (Elementary part) This part consists of 10 elementary Probability and Stat questions. These will be the same (or very similar) questions that are given for Actuarial Exam. Students are expected to successfully complete at least 80% of these problems.

**Part 2.** (Advanced part) This part consists of 3-5 advanced problems from Probability Theory and Math Stat classes. Students are expected to successfully complete at least 60% of these problems.

**Part 3** (Proofs) This part contains 3-5 statements from the __predetermined list__ of about 20 basic well known facts in Probability and Statistics with fairly simple proofs (less than a page). Important: the students are given the list of questions ahead of time and the grading is strict with no partial credits. Students are expected to successfully complete at least 80% of these problems.
**Note: ** the syllabus in any particular section of the Introductory Abstract Algebra, Introductory Analysis, and Mathematical Probablity/Statistics courses might differ slightly from the subject material listed above.

**Sample Algebra Exams**

**Sample Analysis Exams**

**Samples for Probability & Statistics Exam**

**References:**

*Topics in Algebra*, 2nd ed., by Herstein, Chapters 2-5, 6.1-6.3, and 7.

*Algebra*, 3rd ed., by Lang, Chapters 1-6.

*Abstract Algebra*, 3rd ed., by Dummit and Foote, Chapters 1-5, 7-9, and 13-14, excluding 9.6 and 14.9.

*Introduction to Analysis*, by Maxwell Rosenlicht, Chapters 2-7.

*Real Mathematical Analysis*, by Charles Pugh, Chapters 1-4.

*Real Analysis*, 3rd ed., by H.L. Royden, Chapters 3, 4, 5.

*Principles of Mathematical Analysis*, 3rd ed., by Rudin, Chapters 2-8 and 11.

*Measure and Integral*, by Wheeden amd Zygmund, Chapters 3-5.

*A Probability Path* by Resnick

*Probability theory* by Shiryaev

*Measure Theory and Probability Theory* by Athreya and Lahiri

*Mathematical Statistics* by Bickel and Doksum

*Statistical Inference* by Casella

**For information contact:**

Prof. Y. Wang, Graduate Director

Department of Mathematical Sciences

Florida Atlantic University

777 Glades Road

Boca Raton, FL 33431