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, 01 laws, basic inequalities, modes of convergence and their interrelationships, LaplaceStieltjes 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.
CourseSection and Instructors
CourseSection 
Instructor 
Math 768002 
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 # 
9780121741518 
ExtraInfo
Universitywide 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 graduatestudy 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:
 Probability measures on abstract spaces
 Random variables as measurable mappings and their induced distributions
 Independence
 Various modes of convergence especially the fundamental vague/weak convergence
 Weak and strong laws of large numbers; convergence of random series
 Characteristic functions and their application in advanced probability
 Central limit theorems
Course Assessment: Will be based on regular homework, two midterm exams, and one final exam.
Policies
DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official universitywide 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 9735965417 or via email at lyles@njit.edu. 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
1/15 
Introduction 
Measure and Integral 
Week 2
1/22 
Chapter 2 
More measure theory; Probability measures and their distributions 
Week 3
1/29 
Chapter 3 
Random variables, Expectation, Independence 
Week 4
2/5 
Chapter 4 
Various modes of convergence; BorelCantelli lemma 
Week 5
2/12 
Chapter 4 
Vague convergence; examples; equivalent notions; Helly’s extraction principle 
Week 6
2/19 
Chapter 4 
Various criteria for vague/weak convergence 
Week 7
2/26 
MIDTERM EXAM I: MoNday ~ FebrUARY 26, 2018 

Week 8
3/5 
Chapter 4 
Weak convergence; relative compactness; uniform integrability; convergence of moments 
3/12 

Spring recess ( No classes) 
Week 9
3/19 
Chapter 5 
Weak/strong law of large numbers, convergence of random series 
Week 10
3/26 
Chapter 6 
Characteristic functions; properties, uniqueness and inversion 
Week 11
4/2 
Chapter 6 
Characteristic functions; convergence theorems 
Week 12
4/9 
MIDTERM EXAM II:
MONday ~ April 9, 2018 

Week 13
4/16 
Chapter 6 
Characteristic functions; applications 
Week 14
4/23 
Chapter 7 
Liapounov’s central limit theorem 
Week 15
4/30 
Chapter 7 
The LindebergFeller central limit theorem 
Updated by Professor S. Subramanian  1/18/2017
Department of Mathematical Sciences Course Syllabus, Spring 2018