 # Advanced Math Refresher course for PhD students

COURSE OBJECTIVES
This course is designed to serve as an advanced introduction to the mathematical concepts that are used in business research. It is structured to be appropriate for incoming PhD students who have had some quantitative training in the backgrounds, or are planning to focus on the quantitative areas of their disciplines in their PhD programs. The students are expected to be familiar with (but, not proficient in) the concepts covered in the course (see below). Using a combination of lectures, examples, discussions, and exercises, I will devote the class sessions to exploring and articulating the value of mathematical concepts that are useful for doing quantitative business research. I will emphasize:

• Conceptual foundations rather than procedural or computational techniques
• Relating mathematical constructs to business phenomena
Formulation , representation, and interpretation of mathematical models in business research

Here is a sampling of the types of topics we will cover. I may emphasize some topics more than others depending on the background of the students and their needs. Although some of the topics overlap with that of the basic Math camp course taught by Professor Tony Kwasnica, the coverage of the topics in this advanced Math camp will be mathematical in nature.

(1) Real Analysis /Calculus, which includes topics such as real numbers, sequences, limits, and functions, continuity, differentiability, derivatives, maxima and minima, Taylor’s theorem, integration, and multivariate calculus.

(2) Linear Algebra , which includes implicit function theorem, multivariate calculus, vector spaces, matrix operations, eigen systems, and linear and nonlinear optimization.

(3) Probability theory, which includes, sample space and events, Bayes theorem and its applications, conditional independence, random variables, functions of random variables, and stochastic processes.

(4) Statistics, including sampling theory, statistical distributions and their properties, mixing distributions, moment generating functions, estimation, including properties of estimators, hypothesis testing, estimation of linear models, maximum likelihood estimation, and Generalized Method of Moments (GMM) estimators.

(5) Optimization, including brief introductions to linear and nonlinear programming.

TEXTBOOK

There is no textbook for this course, but you may find the following reference books useful. I will distribute any relevant reading materials ahead of class sessions.

1.Marsden, Jerrold E. and Michael J. Hoffman (1993), Elementary Classical Analysis, San Fransisco, CA: W. H. Freeman and Co. (Or any other standard book on real analysis).
2.Rohatgi, Vijay K. and A. K. Md. Ehsanes Saleh (2000), An Introduction to Probability and Statistics, New York, NY: John Wiley & Sons. (Or, any standard book on probability and statistics).
3.Sampit Chatterjee, Ali S. Hadi and Bertram Price (2000), Regression Analysis by Example, 3rd edition, New York: John Wiley & Sons, Inc.
4.William Greene (2003), Econometric Analysis, 5th edition, Englewood Cliffs: Prentice Hall.
5.Luenberger, David G. (2003), Linear and Nonlinear Programming, Second Edition, Springer.
6.Schaum's Outline series such as those for Probability and Statistics (Murray R. Spiegel), Matrix Operations (Richard Bronson), and Operations Research (Richard Bronson).

Course Schedule (Tentative)

 Date Topic Aug 10 (8 - 12pm) Real Analysis/Calculus Aug 12 (12 - 5pm) Probability Theory Aug 14 (8am – 12pm) Linear Algebra Aug 18 (8am – 12Noon) Statistics Aug 20 (9am - 12Noon) Statistics and Optimization

Detailed session plan:

 Session 1 8/10 Ral Analysis/Calculus - Real numbers , functions, sequences, limits - Continuity, differentiability , derivatives - Mean value theorem /Taylor’s theorem, maxima and minima - Integrals, and rules for integration - Multivariate calculus - Nonlinear optimization Session2 8/12 Probability theory - Outcome space, event space, sample space, independence of events, conditional independence, Bayes theorem, hazard function, equally likely outcomes ( permutations and combinations ) - Probability in a continuum, random variables , probability distribution of a random variable, probability distribution of functions of a random variable - Expectation, conditional expectation, and moment generating functions - Joint, conditional, marginal, and sampling distributions Session 3 8/14 Linear Algebra - Vector spaces, basis vectors, matrices, matrix operations, determinants - Inverse matrices, linear dependence, eigen systems , vector and matrix differentiation - The mathematics of least squares curve fitting. Session 4 8/18 Statistics (theory and estimation) - Statistical distributions and their properties -- Binomial, Negative Binomial , Multinomial, Poisson, Exponential, Normal, Beta, Chi-Square, F, Wishart, Gamma, and Dirichlet - Multivariate distributions, mixing distributions - Ordinary Least Squares (OLS) estimation and its properties - Maximum likelihood estimation Session 5 8/20 Statistical estimation and Optimization - Generalized Method of Moments (GMM)) estimation - Introduction to linear programs (if time permits)
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