# 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 18/10 |
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 |

Session28/12 |
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 38/14 |
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 48/18 |
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 58/20 |
Statistical estimation and
Optimization- Generalized Method of Moments (GMM)) estimation - Introduction to linear programs (if time permits) |

Prev | Next |