Course11110415240 lectures
Measure Theoretic Probability 1
Prof. Suprio Bhar · IIT Kanpur
About
This course is aimed at the students who have already learnt about basic Probability distributions and Random variables, and are interested in learning the Mathematical formulation of Probability. We first discuss the modeling of sample space and events into a probability space and, Random variables as functions on such probability spaces. Then, using Measure Theoretic techniques, we discuss a theory of integration which unifies the formulas for the expectations of discrete and absolutely continuous Random variables. Towards the end, we look at various convergence theorems for expectations of Random variables and related inequalities. Students, who complete the course, should be able to apply these results towards advanced studies.
Intended audience: Students who have already learnt about basic Probability distributions and Random variables, and are interested in learning the Mathematical formulation of Probability. PG/Ph.D students and senior UG students are welcome.
Prerequisites: A good background of Real Analysis and Basis Probability Theory (covering Probability distributions and standard Random variables).
Lectures
Week 1: Random Experiments and Sigma-fields
- Lecture 01: Introduction to the course Measure Theoretic Probability 1
- Lecture 02: Sigma-fields and Measurable spaces
- Lecture 03: Fields and Generating sets for Sigma-fields
- Lecture 04: Borel Sigma-field on R and other sets
- Lecture 05: Limits of sequences of sets and Monotone classes
Week 2: Measures and Measure Spaces
- Lecture 06: Measures and Measure Spaces
- Lecture 07: Probability Measures
- Lecture 08: Properties of Measures (Part 1)
- Lecture 09: Properties of Measures (Part 2)
- Lecture 10: Properties of Measures (Part 3)
Week 3: Borel Measurable Functions and Random Variables
- Lecture 11: Measurable functions
- Lecture 12: Borel measurable functions
- Lecture 13: Algebraic properties of Measurable functions
- Lecture 14: Limiting behaviour of measurable functions
- Lecture 15: Random Variables and Random Vectors
Week 4: Law and Distribution Function of a random variable
- Lecture 16: Law or Distribution of an RV
- Lecture 17: Distribution Function of an RV
- Lecture 18: Decomposition of Distribution functions
- Lecture 19: Construction of RVs with a specified law
- Lecture 20: Caratheodery Extension Theorem
Week 5: Correspondence between Distribution functions and Probability measures
- Lecture 21: From Distribution Functions to Probability Measures (Part 1)
- Lecture 22: From Distribution Functions to Probability Measures (Part 2)
- Lecture 23: Lebesgue-Stieltjes Measures
- Lecture 24: Properties of Lebesgue Measure on R
- Lecture 25: Distribution Functions and Probability Measures in higher dimensions
Week 6: Expectation of a discrete RV
- Lecture 26: Integration of measurable functions
- Lecture 27: Properties of Measure Theoretic Integration (Part 1)
- Lecture 28: Properties of Measure Theoretic Integration (Part 2)
- Lecture 29: Monotone Convergence Theorem
- Lecture 30: Computation of Expectation for Discrete RVs
Week 7: Convergence Theorems involving Expectations of RVs
- Lecture 31: MCT and the Linearity of Measure Theoretic Integration
- Lecture 32: Sets of measure zero and Measure Theoretic Integration
- Lecture 33: Fatou's Lemma and Dominated Convergence Theorem
- Lecture 34: Riemann and Lebesgue integration
- Lecture 35: Computations involving Lebesgue Integration
Week 8: Expectation of an Absolutely Continuous RV
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Kanpur
- Department
- Mathematics
- Teacher
- Prof. Suprio Bhar
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed