Probability Foundations for Electrical Engineers

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

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