Lectures

 Lecture 1 Introduction, Probability spaces, properties of probability measures, conditional probability, independence. Lecture 2 Bayes' rule and inference, independent trials, discrete random variables and probability mass functions, cumulative distribution functions, continuous random variables and densities. Lecture 3 Mixed and singular random variables, conditional distributions, functions of random variables, expected values. Lecture 4 Moments, Gaussian random variables, Markov's inequality, Chebyshev's inequality, Chernoff bounds, Moment Generating Functions. Lecture 5 Random vectors, joint cdfs, pdfs and pmfs, marginal distributions, independence, conditional distributions. Lecture 6 Functions of multiple random variables, sums of random variables, characteristic functions, linear transformations of random vectors, expectation and moments of random vectors, correlation and covariance. Lecture 7 Correlation coeeficients, jointly Gaussian random variables, Gaussian Random variables and linear transformations, covariance matrices, jointly Gaussian random variables and conditioning. Lecture 8 More on Gaussian random vectors and conditioning, conditional expectation, iterated expectation, introduction to estimation. Lecture 9 Baysian MMSE estimation, estimation and jointly Gaussian random variables, LLSE estimation. Lecture 10 Discrete-time random processes, i.i.d. processes and laws of large numbers, mean square convergence, convergence in probability. MID-TERM EXAM Lecture 11 Almost sure convergence, convergence in distribution, the central limit theorem. Lecture 12 Finite dimensional distributions and Kolmogorov's theorem, Stationary processes, memoryless processes, stationary increments, independent increments, Markov property, counting processes, random walks. Lecture 13 Markov chains, transition matrices/graphs, n-step transistions, first-step analysis, stationary distributions. Lecture 14 Arrival processes/counting processes, Poisson processes. Lecture 15 Mean and correlation/covariance functions, wide sense stationary processes, Gaussian Processes, Wiener processes. Lecture 16 Multiple random processes, cross correlation functions, Mean-square calculus. Lecture 17 Mean-square integration, random processes and linear systems, power spectral density functions. Lecture 18 Systems driven by white noise; optimal linear filtering; the non-causal Wiener filter; overview of related courses.

A list of lecture topics from 2011 can be found here.