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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, 3^{rd} edition, New York: John Wiley & Sons, Inc.
4.William Greene (2003), Econometric Analysis, 5^{th} 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, ChiSquare, 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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