Rutgers Business School Fall 2016

Introduction to Probability26:960:575

Place: 1 Washington Place, Room 532, Newark, NJ 07102

Time: Wednesdays 10:00-12:50

Instructor: Thomas Lidbetter

Office: 1 Washington Place, Room 1076, Newark, NJ 07102

E-Mail: tlidbetter@business.rutgers.edu

Office Hours: Wednesday 14:30-15:30 or by appointment.

Teaching Material:

� Required Text:

S. M. Ross, A First Course in Probability, 9th Edition (2014).

This introductory text has a very clear exposition, numerous exercises, and applications of probability theory

to everyday problems and situations.

� Recommended Texts:

W. Feller, An Introduction to Probability Theory and Its Applications. Third Edition, J. Wiley & Sons (1967).

R. Weber, Probability.

S. M. Ross, Introduction to Probability and Statistics for Engineers and Scientists, Fifth Edition (2014).

Prerequisites: Graduate students who have finished a basic course in calculus are allowed to take the course.

Grading: Midterm: 30%, Homework 20%, Final Exam: 50%.

Outline of the Course: This is a first course in probability that does not assume any previous knowledge of

probability. It will provide an introduction to advanced mathematical concepts and methods that find extensive use

in many fields of modern Data Science and Operations Research. The course will have a theoretical focus, but theory

will be motivated and illustrated with several examples from areas such as business and engineering. The course will

broadly follow Ross’s ‘A First Course in Probability’, with classes providing intuition and elaboration.

This course is a prerequisite for Stochastic Processes and for Stochastic Calculus for Finance.

Homework: Assignments, given on a weekly/biweekly basis, are to be done individually, unless otherwise stated.

You may discuss the problems with each other, but the work that you submit must be your own. You are expected to

refrain from using solutions from other sources (e.g. previous years’, classes, etc). If you do use outside information,

you must state your sources.

Participation: Classes will be a two-way process, and you will be expected to participate in class discussions and

activities. You may be asked to prepare presentations for classes. I will be available after class for questions, and

will respond to emails promptly.

Course Syllabus:

• Week 1: Introduction

• Week 2 (Ross, Chapter 1): Principles of combinatorial analysis. Examples of combinatorial problems. Permu-

tations and combinations. Stirling’s formula.

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Rutgers Business School Fall 2016

• Week 3 (Ross, Chapter 2): The concept of probability: classical, frequentist and axiomatic definitions.

Inclusion-exclusion formula.

• Week 4 (Ross, Chapter 3): Conditional probability and independence. Law of Total Probability. Bayes’s

Theorem.

• Week 5 (Ross, Chapter 4): Discrete random variables. Examples: Bernoulli, Binomial. Expectation and

variance.

• Week 6 (Ross, Chapter 4): More examples of discrete random variables: Poisson and others. Sums of random

variables. Properties of random variables.

• Week 7 (Ross, Chapter 5): Continuous random variables. Expectation and variance. Examples: Uniform,

Normal, Exponential. Functions of random variables.

• Week 8: MIDTERM EXAM

• Week 9 (Ross, Chapter 6): Jointly distributed random variables. Independent random variables. Conditional

distributions. Buffon’s needle.

• Week 10 (Ross, Chapter 7): Properties of Expectation. Expectation of sums of random variables. Conditional

expectation. Moments.

• Week 11 (Ross, Chapter 8): Convergence and limit theorems. The Markov, Chebyshev and Jensen Inequalities.

The Weak and Strong Laws of Large Numbers, the Central Limit Theorem.

• Week 12 (Weber, Chapter 12) Probability generating functions

• Week 13 (Weber, Chapters 14 and 15) Branching processes. Random walks. Gambler’s ruin.

• Week 14 (Ross, Chapter 9) Simulation, resampling methods.

• Week 15: FINAL EXAM

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