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 |
ExtraInfo
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:
- 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 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 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; Borel--Cantelli 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 Lindeberg-Feller central limit theorem |
Updated by Professor S. Subramanian - 1/18/2017
Department of Mathematical Sciences Course Syllabus, Spring 2018