Department of Statistics and Operations Research

EDWARD CARLSTEIN, Chair

Vladas Pipiras, Director, Program in Mathematical Decision Sciences

Professors

Amarjit Budhiraja, Edward Carlstein, Jason Fine, Joseph Ibrahim, Alan F. Karr, Douglas G. Kelly, Michael Kosorok, Vidyadhar G. Kulkarni, M. Ross Leadbetter, J. Stephen Marron, Andrew Nobel, Vladas Pipiras, J. Scott Provan, Pranab K. Sen, Richard L. Smith, Jayashankar Swaminathan.

Associate Professors

Nilay Argon, Jan Hannig, Chuanshu Ji, Yufeng Liu, Gabor Pataki, Haipeng Shen, Serhan Ziya.

Assistant Professors

Shankar Bhamidi, Shu Lu, Kai Zhang.

Adjunct Professors

Kenneth A. Bollen, Harry L. Hurd, Robert N. Rodriguez.

Lecturer

Charles Dunn.

Professors Emeriti

Charles R. Baker, George S. Fishman, Gopinath Kallianpur, David Rubin, Gordon D. Simons, Walter L. Smith, Shaler Stidham Jr., Jon W. Tolle.

Introduction

The bachelor of science with a major in mathematical decision sciences offers an excellent undergraduate major for students interested in actuarial science, operations research, probability, or statistics, as well as in fields such as business, economics, planning, psychology, and biomedicine where the decision and statistical sciences play an increasingly important role.

Particular areas in which graduates can obtain employment or continue with graduate study include

** Statistics:** Probability and statistics are two of the most frequently applied areas in the mathematical decision sciences. Students in this area study the mathematical theories of probability and statistics and their application to mathematical models that contain an element of uncertainty or randomness. Opportunities for employment are manifold in businesses and government agencies dealing with many branches of the natural and social sciences, including pharmacology, environmental sciences, and many others.

** Operations Research:** In this area, students study mathematical and statistical techniques related to decision making. This branch of the mathematical decision sciences is crucial in business, government, and other management areas where difficult problems that depend on large amounts of data are addressed (for example, complex airline route schedules). In addition to their major courses, students interested in this field are encouraged to take courses in business and economics.

** Actuarial Science:** Actuaries are mathematicians who work primarily in businesses that involve financial risk, including the insurance industry. Students interested in this field take advanced courses in statistics, stochastic processes, and the mathematical theory of risk.

Program of Study

The degree offered is the bachelor of science with a major in mathematical decision sciences. A minor is also available.

Majoring in Mathematical Decision Sciences: Bachelor of Science

Core Requirements

• MATH 547

• STOR 415, 435, 445, 455, and 456

• Four courses from Group A and Group B, including at least two courses from Group A

º Group A: STOR 305, 465, 471, 472, 515, 555, 565

º Group B: BIOL 526; BIOS 664; BUSI 408; COMP 401, 410; MATH 383, 521, 522, 523, 524, 548, 566; STOR 582

Additional Requirements

• COMP 116 (110 may be substituted)

• MATH 231, 232, 233

• STOR 155 and 215

Course Sequencing

In the first two years, students are required to complete the standard calculus sequence as well as introductory courses in statistics, operations research, and computer science. At the beginning of their third year, students take advanced courses in statistics, probability, and operations research. They have a great deal of flexibility in tailoring their program to meet their individual interests.

First and Second Years

All General Education requirements apply. The following courses are required:

• COMP 116 (110 may be substituted)

• MATH 231, 232, 233

• STOR 155*, 215*

Third and Fourth Years

The following courses must be taken by all majors:

• MATH 547

• STOR 415, 435, 445, 455, 456

• Four courses from the following two groups of courses, including at least two from Group A**

º Group A: STOR 305, 465, 471, 472, 515, 555, 565

º Group B: BIOL 526; BIOS 664; BUSI 408; COMP 401, 410; MATH 383, 521, 522, 523, 524, 548, 566; STOR 582

It is recommended that all mathematical decision sciences majors take ECON 101 as a social and behavioral sciences Approaches course. Students interested in the actuarial profession also should take BUSI 100 as a general elective.

*Prospective mathematical decision sciences majors are encouraged to take STOR 155 and 215 as early as possible in their college careers. Each has a prerequisite of MATH 110 or its equivalent and may be taken before, or concurrently with, MATH 231.

**Students wishing to prepare for an actuarial career should include STOR 471, 472, and 555 from Group A in their program and take ECON 410 and 420 and BUSI 408 and 580 as electives. Students who plan to attend graduate school in the mathematical decision sciences (e.g., in operations research or statistics) should include in their program COMP 401, STOR 555, 565, and either MATH 521 or STOR 515.

Mathematical decision sciences majors must complete 123 academic hours. They also must attain at least a grade of C (not C-) in 18 hours of the courses listed under Core Requirements.

Minoring in Mathematical Decision Sciences

Students wishing to minor in the mathematical decision sciences are required to take STOR 155 and 215 as well as three courses from among STOR 305, 415, 435, 445, 455, 456, 465, 471, 472, 515, 555, and 565 that are not counted towards their major requirements.

Honors in Mathematical Decision Sciences

The mathematical decision sciences program offers students the possibility of graduating with honors or highest honors. The requirements for honors can be satisfied in a variety of ways but have a common basis in the minimal standards set by the College of Arts and Sciences. Students interested in graduation with honors should consult the program director prior to the beginning of their senior year.

Advising

All majors and minors have a primary academic advisor in Steele Building. Students are strongly encouraged to meet regularly with their advisor and review their Tar Heel Tracker each semester. After contacting the mathematical decision sciences office (see “Contact Information” below), all majors and minors are also assigned an undergraduate advisor in the department. The department’s undergraduate advisors discuss course planning with current majors and, if needed, minors before registration each semester. The director of undergraduate studies works with prospective majors and minors by appointment. Further information on courses, undergraduate research opportunities, the honors program, careers, and graduate schools may be obtained from the department’s Web site or by contacting the director of undergraduate studies.

Courses for Students from Other Departments

The Department of Statistics and Operations Research offers a variety of courses of potential value to students majoring in other disciplines. Introductory courses include STOR 112, 113, and 215, which are foundation courses in decision models, and the basic statistical courses, STOR 151 and 155. At the intermediate level, STOR 305 provides an introduction to business decision models, while STOR 471 is an introductory course in actuarial science. Substantial coverage of applied statistical methods is provided in STOR 455 and 456. At more advanced mathematical levels, an introduction to probability theory is provided by STOR 435, and the basic theory of statistical inference is given by STOR 555. More advanced deterministic and stochastic models of operations research are provided in STOR 415 and 445.

Special Opportunities in Mathematical Decision Sciences

Departmental Involvement

The mathematical decision sciences program sponsors Carolina’s Actuarial Student Organization (CASO), for students interested in a career in the actuarial sciences. CASO organizes study groups for the actuarial exams, sponsors talks by professional actuaries, keeps members aware of employment opportunities, and maintains contact with alumni and corporations in the field.

Undergraduate Awards

Two undergraduate awards for graduating seniors are given each year by the mathematical decision sciences program. One is the Mathematical Decision Sciences Award, given to the outstanding graduating senior, and the second is the W. Robert Mann Award, given for excellence in actuarial science. Plaques bearing the names of winners are located in the undergraduate study room in Hanes Hall.

Undergraduate Research

Undergraduate research under the direction of faculty members from the Department of Statistics and Operations Research is offered through the independent study and research course, STOR 496. Students pursuing honors are encouraged to contact faculty members about the possibility of enrollment in these courses.

Graduate School and Career Opportunities

Regardless of the electives chosen, the mathematical decision sciences degree program provides excellent preparation for graduate study. Graduates with concentrations in operations research or statistics often continue work at the graduate level in those fields or related areas such as industrial engineering, biostatistics, and environmental science, or enter business school to pursue the master’s in business administration (M.B.A.) degree.

A five-year B.S.–M.S. degree program in operations research is also an option. This program is under the direction of the Department of Statistics and Operations Research. Interested students should consult the program director.

Graduates in the mathematical decision sciences will find numerous opportunities for well-paid, challenging jobs.

Contact Information

Mathematical Decision Sciences Office, CB# 3260, 323 Hanes Hall, (919) 843-6024, kieber@email.unc.edu. Web site: www.stat-or.unc.edu/programs/MDS.

STOR

52 First-Year Seminar: Decisions, Decisions, Decisions (3). In this course, we will investigate the structure of these decision problems, show how they can be solved (at least in principle), and solve some simple problems.

53 First-Year Seminar: Networks: Degrees of Separation and Other Phenomena Relating to Connected Systems (3). Networks, mathematical structures that are composed of nodes and a set of lines joining the nodes, are used to model a wide variety of familiar systems.

56 First-Year Seminar: The Art and Science of Decision Making in War and Peace (3). This seminar will use recently assembled historical material to tell the exciting story of the origins and development of operations research during and after World War II.

60 First-Year Seminar: Statistical Decision-Making Concepts (3). We will study some basic statistical decision-making procedures and the errors and losses they lead to. We will analyze the effects of randomness on decision making using computer experimentation and physical experiments with real random mechanisms like dice, cards, and so on.

61 First-Year Seminar: Statistics for Environmental Change (3). Studies the Environmental Protection Agency’s Criteria Document, mandated by the Clean Air Act; this document reviews current scientific evidence concerning airborne particulate matter. Students learn some of the statistical methods used to assess the connections between air pollution and mortality, and prepare reports on studies covered in the Criteria Document.

62 First-Year Seminar: Probability and Paradoxes (3). The theory of probability, which can be used to model the uncertainty and chance that exist in the real world, often leads to surprising conclusions and seeming paradoxes. We survey and study these, along with other paradoxes and puzzling situations arising in logic, mathematics, and human behavior.

63 First-Year Seminar: Statistics, Biostatistics, and Bioinformatics: An Introduction to the Ongoing Evolution (3). This course is designed to emphasize the motivation, philosophy, and cultivation of statistical reasoning in the interdisciplinary areas of statistical science and bioinformatics.

64 First-Year Seminar: A Random Walk down Wall Street (3). Introduces basic concepts in finance and economics, useful tools for collecting and summarizing financial data, and simple probability models for quantification of market uncertainty.

66 First-Year Seminar: Visualizing Data (3). This seminar looks at a variety of ways in which modern computational tools allow easy and informative viewing of data. Students will also study the kinds of choices that have to be made in data presentation and viewing.

72 First-Year Seminar: Unlocking the Genetic Code (3). Introduces students to the world of genetics and DNA and to the use of computers to organize and understand the complex systems associated with the structure and dynamics of DNA and heredity.

89 First-Year Seminar: Special Topics (3). Special topics course. Content will vary each semester.

112 Decision Models for Business (3). Prerequisite, MATH 110. An introduction to the basic quantitative models of business with linear and nonlinear functions of single and multiple variables. Linear and nonlinear optimization models and decision models under uncertainty will be covered.

113 Decision Models for Economics (3). Prerequisite, MATH 110. An introduction to multivariable quantitative models in economics. Mathematical techniques for formulating and solving optimization and equilibrium problems will be developed, including elementary models under uncertainty.

151 Basic Concepts of Statistics and Data Analysis I (3). Prerequisite, MATH 110. Elementary introduction to statistical reasoning, including sampling, elementary probability, statistical inference, and data analysis. STOR 151 may not be taken for credit by students who have credit for ECON 400 or PSYC 210.

155 Introduction to Statistics (3). Prerequisite, MATH 110. Data analysis; correlation and regression; sampling and experimental design; basic probability (random variables, expected values, normal and binomial distributions); hypothesis testing and confidence intervals for means, proportions, and regression parameters; use of spreadsheet software.

215 Introduction to the Decision Sciences (3). Prerequisite, MATH 110. Introduction to basic concepts and techniques of decision making and information management in business, economics, and the social and physical sciences. Topics include discrete optimization, discrete probability, networks, decision trees, games, Markov chains.

305 Decision Making Using Spreadsheet Models (3). Prerequisite, MATH 152 or STOR 155. The use of mathematics to describe and analyze large-scale decision problems. Situations involving the allocation of resources, making decisions in a competitive environment, and dealing with uncertainty are modeled and solved using suitable software packages.

358 Sample Survey Methodology (BIOS 664) (4). See BIOS 664 for description.

415 Deterministic Models in Operations Research (3). Prerequisite, MATH 547. Linear, integer, nonlinear, and dynamic programming, classical optimization problems, network theory.

435 Introduction to Probability (MATH 535) (3). Prerequisite, MATH 233. Introduction to the mathematical theory of probability, covering random variables; moments; binomial, Poisson, normal and related distributions; generating functions; sums and sequences of random variables; and statistical applications.

445 Stochastic Models in Operations Research (3). Prerequisite, BIOS 660 or STOR 435. Introduction to Markov chains, Poisson processes, continuous-time Markov chains, renewal theory. Applications to queueing systems, inventory, and reliability, with emphasis on systems modeling, design, and control.

455 Statistical Methods I (3). Prerequisite, STOR 155. Review of basic inference; two-sample comparisons; correlation; introduction to matrices; simple and multiple regression (including significance tests, diagnostics, variable selection); analysis of variance; use of statistical software.

456 Statistical Methods II (3). Prerequisite, STOR 455. Topics selected from design of experiments, sample surveys, nonparametrics, time series, multivariate analysis, contingency tables, logistic regression, and simulation. Use of statistical software packages.

465 Simulation Analysis and Design (3). Prerequisite, STOR 445. Introduces concepts of random number generation, random variate generation, and discrete event simulation of stochastic systems. Students perform simulation experiments using standard simulation software.

471 Long-Term Actuarial Models (3). Prerequisite, STOR 435. Probability models for long-term insurance and pension systems that involve future contingent payments and failure-time random variables. Introduction to survival distributions and measures of interest and annuities-certain.

472 Short Term Actuarial Models (3). Prerequisite, STOR 435. Short term probability models for potential losses and their applications to both traditional insurance systems and conventional business decisions. Introduction to stochastic process models of solvency requirements.

496 Undergraduate Reading and Research in Statistics and Operations Research (1–3). Permission of the director of undergraduate studies. This course is intended mainly for students working on honors projects. May be repeated for credit.

515 Computational Mathematics for Decision Sciences (3). Permission of the instructor. Reviews basic mathematical and computational theory required for analyzing models that arise in operations research, management science, and other policy sciences. Solution techniques that integrate existing software into student-written computer programs will be emphasized.

555 Mathematical Statistics (3). Prerequisite, STOR 435. Functions of random samples and their probability distributions, introductory theory of point and interval estimation and hypothesis testing, elementary decision theory.

565 Introduction to Machine Learning (3). Prerequisites, STOR 215 or MATH 381, and STOR 435. Introduction to theory and methods of machine learning including classification; Bayes risk/rule, linear discriminant analysis, logistic regression, nearest neighbors, and support vector machines; clustering algorithms; overfitting, estimation error, cross validation.

582 Neural Network Models for the Decision and Cognitive Sciences (3). Prerequisite, MATH 231, PHIL 155, PSYC 210, or STOR 155 or 215. The interactions between cognitive science and the decision sciences are explored via neural networks. The history of these networks in neuroscience is reviewed and their adaptation to other fields such as psychology, linguistics, and operations research is presented.

612 Models in Operations Research (3). Required preparation, calculus of several variables, linear or matrix algebra. Formulation, solution techniques, and sensitivity analysis for optimization problems which can be modeled as linear, integer, network flow, and dynamic programs. Use of software packages to solve linear, integer, and network problems.

614 Linear Programming (3). Required preparation, calculus of several variables, linear or matrix algebra. The theory of linear programming, computational methods for solving linear programs, and an introduction to nonlinear and integer programming. Basic optimality conditions, convexity, duality, sensitivity analysis, cutting planes, and Karush-Kuhn-Tucker conditions.

634 Measure and Integration (3). Required preparation, advanced calculus. Lebesgue and abstract measure and integration, convergence theorems, differentiation. Radon-Nikodym theorem, product measures. Fubini theorems. Lp spaces.

635 Probability (MATH 635) (3). Prerequisite, STOR 634. Permission of the instructor for students lacking the prerequisite. Foundations of probability. Basic classical theorems. Modes of probabilistic convergence. Central limit problem. Generating functions, characteristic functions. Conditional probability and expectation.

641 Stochastic Models in Operations Research I (3). Prerequisite, STOR 435. Review of probability, conditional probability, expectations, transforms, generating functions, special distributions, and functions of random variables. Introduction to stochastic processes. Discrete-time Markov chains. Transient and limiting behavior. First passage times.

642 Stochastic Models in Operations Research II (3). Prerequisite, STOR 641. Exponential distribution and Poisson process. Birth-death processes, continuous-time Markov chains. Transient and limiting behavior. Applications to elementary queueing theory. Renewal processes and regenerative processes.

654 Statistical Theory I (3). Required preparation, two semesters of advanced calculus. Probability spaces. Random variables, distributions, expectation. Conditioning. Generating functions. Limit theorems: LLN, CLT, Slutsky, delta-method, big-O in probability. Inequalities. Distribution theory: normal, chi-squared, beta, gamma, Cauchy, other multivariate distributions. Distribution theory for linear models.

655 Statistical Theory II (3). Prerequisite, STOR 654. Point estimation. Hypothesis testing and confidence sets. Contingency tables, nonparametric goodness-of-fit. Linear model optimality theory: BLUE, MVU, MLE. Multivariate tests. Introduction to decision theory and Bayesian inference.

664 Applied Statistics I (3). Permission of the instructor. Basics of linear models: matrix formulation, least squares, tests. Computing environments: SAS, MATLAB, S+. Visualization: histograms, scatterplots, smoothing, QQ plots. Transformations: log, Box-Cox, etc. Diagnostics and model selection.

665 Applied Statistics II (3). Prerequisite, STOR 664. Permission of the instructor for students lacking the prerequisite. ANOVA (including nested and crossed models, multiple comparisons). GLM basics: exponential families, link functions, likelihood, quasi-likelihood, conditional likelihood. Numerical analysis: numerical linear algebra, optimization; GLM diagnostics. Simulation: transformation, rejection, Gibbs sampler.