## Lectures

 Topics Reading Lecture 1 Introduction, Probability spaces, Sigma Algebras, Properties of probability measures, Sect. 1.1-1.2 Lecture 2 Conditional probability, Statistical independence, conditional independence, Repeated trials, random variables, CDFs, PMFs, PDFs, mixed and singular random variables. Sect. 1.3.1-1.3.3 Lecture 3 Multiple random variables, joint distributions, stochastic processes, the Bernoulli process. Sect. 1.3.4 -1.4 Lecture 4 Expected values, moments, sums of random variables, conditional expectation, The coupon collector problem. Sect. 1.5 Lecture 5 Markov's inequality, Chebyshev's inequality, Chernov Bounds. Sect. 1.6 Lecture 6 Moment generating functions and sums of independent random variables, log-moment generating functions, Convergence of random variables: mean-squared convergence, convergence in probability, the weak law of large numbers. Sect. 1.6-1.7 Lecture 7 Almost sure convergence, the strong law of large numbers, convergence in distribution,, the central limit theorem. Sect. 1.7 & 5.2 Lecture 8 Central limit theorem examples, Characteristic functions, Counting Processes and the Poisson Process, memoryless property, fresh-restart property. Sect. 2.1-2.2 Lecture 9 More on Poisson Processes: increment properties, alternative definitions, Splitting and combining Poisson Processes. Sect. 2.2-2.3 Lecture 10 Markov Chains: definitions, transition Matrices/Graphs, first-step analysis. Notes Lecture 11 MID-TERM EXAM Lecture 12 Markov Chains: Stationary distributions, state classifications, recurrence, periodicity. Sect. 4.1-4.2 & 6.2 Lecture 13 Markov Chains: ergodic chains, convergence to stationary distributions, balance equations. Sect. 4.2-4.3, 6.1-6.2 Lecture 14 Gaussian random vectors: linear transformations, moment generating functions. Sect. 3.1 -3.3 Lecture 15 Gaussian random vectors: joint probability distributions and properties of covariance matrices Sect. 3.3 -3.4 Lecture 16 Conditioning and Gaussian random vectors; Introduction to estimation Sect. 3.5, Sect. 10.1. Lecture 17 Estimation and Gaussian random vectors, linear estimation, Intro. to Gaussian Processes. Sect. 10.2-10.3. Lecture 18 Stationary processes, Properties of covariance functions, Wiener processes. Sect. 3.6.1, 3.6.9 Lecture 19 More on Wiener prcess, random processes and linear systems, spectral density.

A list of lecture topics from 2017 can be found here.