## Lectures from 2010

 Lecture 1 Introduction, Probability spaces. Lecture 2 Properties of probability measures, conditional probabilities, independence. Lecture 3 Independent trials, combinations, permutations. Lecture 4 Random variables, distributions, discrete Random variables, p.m.f.'s Lecture 5 Continuous random variables, densities, mixed random variables, examples of common distributions. Lecture 6 Gaussian random variables, conditional distributions, functions of random variables. Lecture 7 Expected values, moments. Lecture 8 Markov inequality, Chebyshev's inequality, Chernoff bounds, moment generating functions, Characteristic functions. Lecture 9 Introduction to random vectors, joint distributions, joint densities, marginals. Lecture 10 More on Random vectors: independence, moments, correlation. Lecture 11 Two jointly Gaussian random variables, functions of random vectors. Lecture 12 Linear transformations of random vectors, condition distributions. Lecture 13 More on conditional distributions, conditional expectation, MMSE estimation. Lecture 14 More on MMSE estimation; scalar Gaussian case. Lecture 15 MMSE estimation - vector case; jointly Gaussian Random variables (N>2). Lecture 16 More on Jointly Gaussian random variables and MMSE estimation; covariance matrices. Lecture 17 Laws of large numbers, mean-square convergence, convergence in probability. Lecture 18 Almost sure convergence; strong law of large numbers; convergence in distribution; the central limit theorem. Lecture 19 More on the central limit theorem; random processes. Lecture 20 Discrete-time Random processes: Bernoulli processes, Binomial counting process, simple random walk; stationarity, Memoryless and Markov properties, independent and stationary increments. Lecture 21 Random walk on a graph; Poisson processes. Lecture 22 More on Poisson processes; random telegraph signals. Lecture 23 Formally specifying random processes, Kolmorogorov's consistency conditions; mean and correlation/covariance functions; wide sense stationarity. Lecture 24 Properties of covariance functions; Brownian motion. Lecture 25 Mean-squared continuity, mean-squared derivatives. Lecture 26 Mean-squared integration; random processes and linear systems. Lecture 27 Spectral analysis for random processes; application to linear systems. Lecture 28 Introduction to optimal filtering; overview of related courses.