Course10810608396 lectures
Probability Foundations for Electrical Engineers
Prof. Krishna Jagannathan · IIT Madras
About
This is a graduate level class on probability theory, geared towards students who are interested in a rigorous development of the subject. It is likely to be useful for students specializing in communications, networks, signal processing, stochastic control, machine learning, and related areas. In general, the course is not so much about computing probabilities, expectations, densities etc. Instead, we will focus on the ‘nuts and bolts’ of probability theory and aim to develop a more intricate understanding of the subject. For example, emphasis will be placed on deriving and proving fundamental results, starting from the basic axioms.
INTENDED AUDIENCE : M.Tech/M.S/PhD students, who plan to specialize in communications, networks, signal processing, stochastic control, machine learning, or related areas.
PREREQUISITES : There will be no official pre-requisites. Although the course will build up from the basics, it will be taught at a fairly sophisticated level. Familiarity with concepts from real analysis will also be useful. Perhaps the most important prerequisite for this class is an intangible one, namely mathematical maturity.
Lectures
Week 1
- Introduction
- Cardinality
- Countability
- Uncountable sets-1
- Uncountable sets-2
- Probability spaces-Introduction
- Probability spaces-Algebra
- Probability spaces-σ-algebra
Week 2
- Probability spaces-Measurable space
- Properties of probability measures
- Continuity of probability measure
- Discrete probability space-finite and countably infinite sample space
- Discrete probability space-Uncountable sample space
- Generated σ-algebra, Borel Sets
- Borel sets
- Uniform probability measure on Borel sets-Lebesgue measure
Week 3
- Carathéodory’s extension theorem
- Lebesgue measure (contd)
- Infinite coin toss model
- Infinite coin toss model (Contd)
- Conditional probability
- Properties of conditional probability
- Independence of events
- Independence of σ-algebras
Week 4
- Borel-Cantelli Lemma 1
- Borel-Cantelli Lemma 2
- Random Variables
- Random Variables (Contd)
- Cumulative Distribution Function
- Properties of CDF
- Types of Random Variables
- Examples of Random Variables
Week 5
- Continuous Random Variables - 1
- Examples of Continuous Random Variables - 1
- Continuous Random Variables - 2,Examples of Continuous RandomVariables - 2
- Singular Random Variables
- Several Random Variables - 1
- Several Random Variables - 2
- Independent Random Variables - 1
- Indiependent Random Variables - 2
Week 6
- Conditional PMF, Jointly Continuous Random Variables - 1
- Jointly Continous RandomVariables - 2
- Jointly Continuous RandomVariables - 3
- Conditional CDF
- Transformation of RandomVariables - 1
- Transformation of RandomVariables - 2; IndependentRandom Variables
- Sums of Discrete RandomVariables
- Sums of Jointly ContinuousRandom Variables
Week 7
- Sums of Random Number of Random Variables
- General Transformations of Random Variables
- Jacobian Formula
- Examples Illustrating the use of Jacobian Formula
- Introduction Integral and Expectation
- Definition of the Abstract Integral
- Simple Functions
- Computing Expectation using Simple Functions, Properties of Integrals
Week 8
- Properties of Integrals (Contd.)
- Inclusion Exclusion Formula using Indicator RVs and Expectation
- Monotone Convergence Theorem - 1
- Monotone Convergence Theorem - 2
- Expectation of a Discrete Random Variable
- Examples of Expectation of Discrete Random Variables
- Expectation of Function of Random Variable
- Some Examples of Computing Expectation
Week 9
- Fatou’s Lemma
- Dominated Convergence Theorem
- Variance
- Covariance
- Covariance Correlation Coefficient - 1
- Covariance Correlation Coefficient - 2
- Conditional Expectation
- Properties of Conditional Expectation
Week 10
- MMSE Estimator
- Transforms
- Moment Generating Function - 1
- Moment Generating Function - 2
- Characteristic Function - 1
- Characteristic Function - 2
- Characteristic Function - 3
- Characteristic Function - 4
Week 11
- Concentration Inequalities - 1
- Concentration Inequalities - 2
- Convergence of Random Variables - 1
- Convergence of Random Variables - 2
- Convergence of Random Variables - 3
- Convergence of Random Variables - 4
- Convergence of Random Variables - 5
- Convergence of Random Variables - 6
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Madras
- Department
- Electrical Engineering
- Teacher
- Prof. Krishna Jagannathan
- Level
- Undergraduate
- Language
- English
- Subjects
- Engineering
- 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