INDIAN STATISTICAL INSTITUTE

(Delhi Centre)

7 S. J. S. Sansanwal Marg, New Delhi 110016

Probability Theory

M.Stat. Ist Year (NB-Stream)

Academic Year 2022 - 2023: Semester I

Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office: Room # 208 on the first floor of the Faculty Block.

Class Time: Monday and Thursday 09:15 AM - 11:15 AM (2 x 120 minutes per week).

Lecture Hall: Room # 23 Academic Block (First Floor).

Google Classroom for all announcements and other information purpose.

Course Duration (including examinations): August 01 - November 30, 2022
(Total of 18 Weeks = 7 Weeks of Classes + 1 Midterm Examination Week + 7 Weeks of Classes + 1 week of Study Break + 2 Final Examination Weeks).

Midterm Examination:
Date: September 20, 2022
Duration: 10:00 AM - 12:00 noon (Two hours)
Venue: Conference Hall, First Floor, Administrative Block

Final Examination:
Date: November 25, 2022
Duration: 10:00 AM - 01:00 PM (Three hours)
Venue: TBA

Course Outline:

• Basic Probability (approximately 8 to 9 weeks):
  • Distribution functions on real line. Probability distributions on real line, existence and uniqueness.
  • Concepts of probability space, random variable, distribution. Discrete and continuous random variables.
  • Concept of product space, independent random variables.
  • General definition of expected value. Properties of expectation, linearity, order-preserving property. Monotone Convergence Theorem (MCT), Fatou's Lemma and Dominated Convergence Theorem (DCT). Expected value for discrete and continuous random variables and their functions. Expectation of product for independent random variables. Hölder, Minkowski and Jensen's inequalities.
  • Different modes of convergence of random variables and their relations. First and Second Borel-Cantelli Lemmas. Markov and Chebyshev inequalities. Weak Law of Large Numbers (WLLN), Strong Law of Large Numbers (SLLN).
  • Weak convergence, definition and examples. Representation theorem, Continuous Mapping Principle, Convergence Together Lemma (Slutsky's Theorem). Fundamental Theorem of Statistics (Glivenko-Cantelli Lemma).
  • Characteristic functions, definition, examples. Moment expansion, characteristics function for Normal distribution. Inversion formula, uniqueness theorem, density formula (statement only). Characteristic function of Cauchy and Double Exponential distributions. Characteristic functions and weak convergence, Lévy's Continuity Theorem. Central Limit Theorem (CLT) for i.i.d. finite variance case. Lindeberg & Lyaponov CLTs (statements only).
• Markov Chains and Stochastic Processes (approximately 5 to 6 weeks):
  • Discrete Markov chains with countable state space, Examples including two state chain, SSRW, random walk, gambler's ruin, birth and death chain, renewal chain, Ehrenfest chain, card shuffling and branching processes.
  • Classification of states, recurrence and transience; absorbing states, irreducibility, decomposition of state space into irreducible classes, examples.
  • Stationary distributions, limit theorems, positive and null recurrence.
  • Definition of periodicity, periodic and aperiodic chains.
  • Limit theorems for aperiodic irreducible chains.
  • Branching processes: introduction, first and last progeny decomposition, generating function, survival probability. Definitions and basic introduction to the Sub-critical, critical and super-critical regimes.

References:

• Basic Probability:
  • An Introduction to Probability Theory and Its Applications (Vol - I & II) by William. Feller.
  • Introduction to Probability Theory by Paul G. Hoel, Sidney C. Port and Charles J. Stone.
  • A First Course in Probability by S. M. Ross.
  • Elementary Probability Theory by Kai L. Chung.
  • Probability An Introduction by Geoffrey Grimmett and Dominic Welsh.
  • Probability by Jim Pitman.
  • Basic Probability Theory by R. B. Ash.
  • Probability and Measure by P. Billingsley.
  • Probability: Theory and Examples by R. Durrett.
• Markov Chains and Stochastic Processes:
  • Introduction to Stochastic Processes by Paul G. Hoel, Sidney C. Port and Charles J. Stone.
  • An Introduction to Probability Theory and Its Applications (Vol - I & II) by William Feller.
  • Probability An Introduction by Geoffrey Grimmett and Dominic Welsh.
  • Probability: Theory and Examples by Rick Durrett (ONLY Chapter 5).

Prerequisites:

• Real Analysis (at the level of Principles of Mathematical Analysis by W. Rudin).
• Liner Algebra (at the level of Finite Dimensional Vector Spaces by P. R. Halmos).

Grading Policy:

• Assignments: 0% of the total credit.
• Quizzes: 20% of the total credit.
• Midterm Exam: 30% of the total credit.
• Final Exam: 50% of the total credit.

Assignment Policies:

• There will be a total of 14 sets of homework assignments.
• The assignments will be given in the Google Classroom on every Thursday, starting from August 04, 2022.
• Assignments will NOT be graded, but they will be used for the in class surprise tests/quizzes.

Quiz Policies:

• There will be four or more quizzes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover materials done in the lectures given in the weeks prior to it.
• Each quiz will be of 20 points and will be of 30 minutes duration.
• Final grade for the quizzes will be (Best Score Before Midterm + Best Score Between Midterm and Final)/2.
• There will be NO supplementary quiz given for any student who may miss a quiz for whatsoever reason. If you miss one then do not worry, try doing well in the others.
• All quizzes will be part of the final grading.
• All quizzes will be closed note and closed book examinations.

Exam Policies:

• The Midterm and the Final Examinations will be open notes examinations. That means, students will be allowed to bring his/her own hand written notes, study materials, list of theorems etc. But no printed or photocopied materials will be allowed.
• Any unfair means used by any students in the examinations will be dealt with the strictest possible measures, as per the Institute rules. In particular, if any student is found to be using any kind of unfair practice during any of the examinations (including the quizzes) then he/she will be awarded ZERO in that examination.

Regrading Policy:

• Regrading of homeworks or exams will only be undertaken in cases where, you believe there has been a genuine error or misunderstanding. Please note that our primary aim in grading is consistency, so that all students are treated the same; for this reason, we will not adjust the score of one student on an issue of partial credit, unless the score allocated clearly deviates from the grading policy we adopted for that problem.
• If you wish to request a regrading of a homework or exam, you must return it to the instructor with a written note on a separate piece of paper explaining the problem.
• The entire assignment or the exam may be regraded, so be sure to check the solutions to ensure that your overall score will go up after regrading.
• All such requests must be received within one week from the date on which the homework or exam was made available for return.

Last modified November 08, 2022.