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Week 1: Introduction, Cardinality and Countability, Probability Space
Week 2: Properties of Probability Space, Discrete Probability Space, Generated \sigma-algebra
Week 3: Borel sets, Caratheodory’s extension theorem, Lebesgue Measure, Infinite coin toss model
Week 4: Conditional probability, Independence, Borel-Cantelli Lemmas
Week 5: Random variables, Distribution function, Types of random variables
Week 6: Discrete Random variables, Continuous random variables, Singular random variables
Week 7: Several random variables, joint distribution, independent random variables
Week 8: Transformation of random variables
Week 9: Integration and Expectation, properties of integrals, Monotone convergence, Dominated convergence, Expectation over different spaces
Week 10:Variance, covariance, and conditional expectation
Week 11:Transform techniques: moment generating function, characteristic function
Week 12:Convergence of random variables, Laws of large numbers, Central limit theorem
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.INDUSTRIES SUPPORT :Research labs
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