Advanced Probability Theory

By Prof. Niladri Chatterjee   |   IIT Delhi
Learners enrolled: 2521
The course introduces the concept of probability through Kolmogorov’s Axioms. It develops the concept of probability density function, cumulative distribution function, and introduces the concept of a random variable. Different theoretical probability distributions, both discrete and continuous are introduced, and their properties are studied. Different generating functions, viz. MGF, PGF, and characteristic functions of different variables are discussed. It also introduces functions of one or two random variables, and derived random variables, such as t, chisq, F are studied in details. The concept of Order Statistics is introduced, and derivation of distribution of range, median etc. are discussed. Different limit theorems are discussed in detail.

INTENDED AUDIENCE : Undergraduate and postgraduate students in statistics, mathematics, and Machine learning
PREREQUISITES : Basics of Real Analysis, Functions of two variables, Convergence of a function.
INDUSTRY SUPPORT : Most Financial companies
Course Status : Completed
Course Type : Core
Duration : 12 weeks
Category :
  • Mathematics
Credit Points : 3
Level : Undergraduate/Postgraduate
Start Date : 24 Jan 2022
End Date : 15 Apr 2022
Enrollment Ends : 07 Feb 2022
Exam Date : 24 Apr 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1 : Introduction, Sample Space, Probability Axioms,  Theorems on Union and Intersections of events in a Sample Spaces.  Bertrand’s Paradox.
Week 2 : Conditional Probability, Bayes Theorem,  Probability on Finite Sample Spaces. Independence of Events..
Week 3 : Introduction to Random variables – discrete & continuous  Random variables Discrete random variables - Uniform, Bernoulli, Binomial, Geometric, Poisson Distributions, Hypergeometric, Negative Binomial
Week 4 :
 Continuous Random variables: Uniform, Normal, Exponential, Gamma, Cauchy, Beta1 and Beta2
Week 5 : Moments of a distribution, Bivariate distribution, Covarience and Correlation
Week 6 : Generating Functions and their properties: Moment Generating Function Characteristic Functions and Probability Generating Function
Week 7 : Poisson Process,  Conditional Expectations and Variance, Chebyshev's Inequality and Introduction to Bivariate Normal.
Week 8 : Functions of Random Variables, Introduction to t and F distribution.
Week 9 : Order Statistics
Week 10 : Limit Theorems: Mode of Convergence
Week 11 : Laws of Large numbers
Week 12 : Central Limit Theorems

Books and references

An Introduction to Probability and Statistics.  Vijay K Rohatgi and A.K.Md. Ehsanes Saleh,   Wiley.

Mathematical Statistics.  Suddhendu Biswas and G.L. Sriwastav. Narosa

Instructor bio

Prof. Niladri Chatterjee

IIT Delhi
Prof. Niladri Chatterjee is a professor of Department of Mathematics, IIT Delhi. He is B.Stat and M.Stat from Indian Statistical Institute Kolkata. He is an M.Tech in Computer Science and PhD in Computer science from University College London. His major research interests are Statistical Modeling, Big Data, Artificial Intelligence, Machine Learning, Natural Language Processing among others. He has more than 30 years of research and teaching experience. He is also member of several Government committees related to data science. He is currently Chair Professor in AI and Machine Learning.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 24 April 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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