INDIAN STATISTICAL INSTITUTE (Delhi Centre) 7 S. J. S. Sansanwal Marg, New Delhi 110016

Instructor: Antar Bandyopadhyay (Email: antar (at) isid (dot) ac (dot) in Office: Room # 208 First Floor of Faculty Block, ISI, Delhi).

Class Time: Monday & Wednesday 10:30 AM - 12:00 PM (at the BSDS New Classroom, First Floor, Indian Bank Building, ISI, Delhi & Online/Hybrid in the other locations).

Tutorial Times: Friday 10:30 AM - 12:00 PM.

Lecture Hall: BSDS New Classroom, First Floor, Indian Bank Building, ISI, Delhi and at the respective BSDS Classrooms in Kolkata & Bangalore. (Note: For August 11 - 20, 2025 for Delhi the Classroom will be Seminar Room II on the First Floor of the Administrative Block)

Google Classroom for all announcements and other information purpose.

TAs:

• At ISI, Kolkata: Aritra Majumdar, SRF, Statistics, TSMU, Kolkata (E-Mail: aritram425 (at) gmail (dot) com ).
• At ISI, Delhi: Deborshi Das, JRF, Statistics, TSMU, Delhi (E-Mail: deborshidas6 (at) gmail (dot) com> ).
• At ISI, Bangalore: Dibyendu Das, SRF, Mathematics, TSMU, Bangalore (E-Mail: >rs_math2202 (at) isibang (dot) ac (dot) in ).

Course Duration: August 11, 2025 - November 28, 2025 (total of 14 weeks classes for which 7 weeks will be prior to the midterm week and 7 weeks after).

Midterm Examination: Exact date TBA (during October 06 - 11, 2025)
Note: There is an week of break before Midterm (September 29 - October 05, 2025 due to the Durga Puja Celebration)

Final Examination: Exact date TBA (during December 08 - 17, 2025)
Note: There is an week of study break before Final (December 01 - 07, 2025)

Backpaper Examination: Exact date TBA (during the week of January 08 - 15, 2026)

Course Outline:

Pre-Midterm:

• Bivariate Normal distribution, density function and basic properties. Marginal and conditional distributions. Spherical symmetry of the Bivariate Normal distribution. (pending from Probability I) [3 lectures]
• Distributions of functions of random vectors and Jacobian formula. Examples. [2 lectures]
• Sampling distribution for mean and sample variance of i.i.d. Normal observations. [2 lectures]
• Cauchy-Schwartz, Markov and Chebyshev inequalities. Weak Law of Large Numbers (WLLN) and Strong Law of Large Numbers (SLLN) (statements only). [3 lectures]
• Concept of almost sure and in probability convergence. Concept of “infinitely often”, First and Second Borel-Cantelli Lemmas. [2 lectures]
  
  Post-Midterm:
  
• Concepts of L₁, L₂ and general L_p convergence where p > 1. [2 lectures]
• Statements of the Monotone Convergence Theorem (MCT), Fatou’s Lemma and the Dominated Convergence Theorem (DCT). [3 lectures]
• Basic notion of distributional convergence. Statement of the i.i.d. Central Limit Theorem (CLT). Slutsky’s Theorem. Delta method. Multivariate CLT, Cramer-Wold device, statement and examples. [3 lectures]
• Basic idea of generating functions. Definition and examples of probability generating function (pgf) for discrete integer valued random variables, moment generating functions (mgf) and the characteristic function (chf). Properties of characteristic functions, uniqueness, inversion and density formula (statements only). Connection between chf and distributional convergence, Levy’s Continuity Theorem (statement only). [4 lecrures]

Knowledge requirement: Mathematics I, II & Probability I of BSDS syllabus

Text :Jim Pitman - Probability.

Other References : Here are two more books which are good to look at for reading and problem solving. You may not want to buy them, best would be to borrow these from the library.

• R. G. Hoel, S. C. Port and C. J. Stone - Introduction to Probability Theory.
• Sheldon M. Ross - A First Course in Probability, 6th Edition.

Grading Policy:

• Assignments: 10% of the total credit.
• Quizzes: 10% 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 each carrying a total of 10 points. The average of the 10 best assignment scores will be taken for the Assignment component of the final grading.
• The assignments will be uploaded in the Google Classroom on every Wednesday, starting from August 13, 2025. Each assignment will be due on the following Friday in the tutorial. You need to submit your handwritten hard copy of your solution of the assignment to your TA in the tutorial. For example the first assignment is due on Friday, August 22, 2025.
• Each Assignment will nominally be based on the course materials which will be covered in the class in the week of its assignment. Occasionally an assignment may be on a material covered in the previous weeks and/or may introduce new concepts to complement the lectures.
• Late submission of an assignment will NOT BE ACCEPTED. If you can not submit an assignment on time, don't worry about it and try to do well in the others. It will not count in your final grade since you have four extra assignments anyway.
• Assignments will be graded by the respective TAs at your own location and the graded assignments will be returned to you in the tutorials locally. Please note only a few problems will be graded by the TAs randomly selected from the assignments. The problems graded by the TAs at different locations may be different. However, problems graded in one location will be same for all students in that Centre.

Quiz Policies:

• There will be at least four 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 10 points and will be of 20 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 close notes, close book examinations. That means, students will NOT be allowed to bring his/her own hand written notes, study materials, list of theorems etc., any printed or photocopied materials including books and handouts.
• 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 homework 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 August 10, 2025.