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Math 768: Probability Theory
Spring 2018 Graduate Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

Course Information

Course Description: Measure theoretic introduction to axiomatic probability. Probability measures on abstract spaces and integration. Random variables and distribution functions, independence, 0-1 laws, basic inequalities, modes of convergence and their interrelationships, Laplace-Stieltjes transforms and characteristic functions, weak and strong laws of large numbers, conditional expectation, discrete time martingales. Effective From: Spring 2009.

Number of Credits: 3

Prerequisites: Math 645 or departmental approval.

Course-Section and Instructors

Course-Section Instructor
Math 768-002 Professor S. Subramanian

Office Hours for All Math Instructors: Spring 2018 Office Hours and Emails

Required Textbooks:

Title A Course in Probability Theory
Author Kai Lai Chung
Edition 2nd
Publisher Academic Press
ISBN # 978-0121741518

University-wide Withdrawal Date:The last day to withdraw with a w is Monday, April 2, 2018. It will be strictly enforced.

Course Goals

Course Objectives: This course will follow the philosophy of the eminent Kai Lai Chung, providing a sustained tour of rigorous probability. Topics include measure and integral, probability measures on abstract spaces, random variables, induced measures and distributions, mathematical expectation, independence, basic probability inequalities, convergence concepts and their interrelationships, characteristic functions, weak and strong laws of large numbers, and the central limit theorem. The aim is to prepare the student for graduate-study readiness in further advanced courses/topics such as weak convergence in function spaces, statistical large sample theory, random walk, Markov processes and Martingales.

Course Outcomes: On successful completion, students will be able to demonstrate understanding and analytical problem solving in the following topics:

Course Assessment: Will be based on regular homework, two midterm exams, and one final exam.


DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

Homework 25%
Midterm Exams 40%
Final Exam 35%

Your final letter grade will be based on the following tentative curve.

A 86 - 100 C+ 71 - 75
B+ 81 - 85 C 66 - 70
B 76 - 80 F 0 - 65

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced.

Homework Policy: Homework assignments are due within a week unless announced otherwise by instructor.   Late homework will not be accepted.

Exams: There will be two midterm exams held in class during the semester and one comprehensive final exam. Exams are held on the following days:

Midterm Exam I February 26, 2018
Midterm Exam II April 9, 2018
Final Exam Period May 4 - 10, 2018

The final exam will test your knowledge of all the course material taught in the entire course. Make sure you read and fully understand the Math Department's Examination Policy. This policy will be strictly enforced.

Makeup Exam Policy: To properly report your absence from a midterm or final exam, please review and follow the required steps under the DMS Examination Policy found here:

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

Additional Resources

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at The office is located in Fenster Hall, Room 260.  A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Spring 2018 Academic Calendar, Registrar)

Date Day Event
January 16, 2018 T First Day of Classes
January 22, 2018 M Last Day to Add/Drop Classes
March 11 - 18, 2018 Su - Su Spring Recess - No Classes/ University Closed
March 30, 2018 F Good Friday - No Classes/ University Closed
April 2, 2018 M Last Day to Withdraw
May 1, 2018 T Friday Classes Meet - Last Day of Classes
May 2 - 3, 2018 W - R Reading Days
May 4 - 10, 2018 F - R Final Exam Period

Course Outline

Week Section Topic
Week  1
Introduction Measure and Integral
Week 2
Chapter 2 More measure theory; Probability measures and their distributions
Week 3
Chapter 3 Random variables, Expectation, Independence
Week 4
Chapter 4 Various modes of convergence; Borel--Cantelli lemma
Week 5
Chapter 4 Vague convergence; examples; equivalent notions; Helly’s extraction principle
Week 6
Chapter 4 Various criteria for vague/weak convergence
Week 7
MIDTERM EXAM I:  MoNday ~ FebrUARY 26, 2018
Week 8
Chapter 4 Weak convergence; relative compactness;  uniform integrability; convergence of moments
3/12 Spring recess ( No classes)
Week 9
Chapter 5 Weak/strong law of large numbers, convergence of random series
Week 10
Chapter 6 Characteristic functions; properties, uniqueness and inversion
Week 11
Chapter 6 Characteristic functions; convergence theorems            
Week 12
MONday ~ April 9, 2018
Week 13
Chapter 6 Characteristic functions; applications
Week 14
Chapter 7 Liapounov’s  central  limit theorem
Week 15
Chapter 7 The Lindeberg-Feller  central  limit theorem

Updated by Professor S. Subramanian - 1/18/2017
Department of Mathematical Sciences Course Syllabus, Spring 2018