## The subject follows the following book closely:

·         [LeGar08] A. Leon-Garcia, Probability, Statistics and Random Processes for Electrical Engineering, 3rd Edition 2008. (Previous editions can also be used).

## Complementary Literature:

·

·         [Ro10]  S. M. Ross, Introduction to Probability Models, 10th Edition, 2010. (Previous editions are also OK).

·

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·         [Pap91] A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd Edtion, 1991. (Previous editions are also OK).

·         [Dur99] R. Durret, Essentials of Stochastic Processes, 1999.

·         [DeGr12]  M. H. DeGroot, M. J. Schervish, Probability and Statistics, 4th Edition, 2012. (Previous editions are also OK).

 Section Outline [LeGar08] Sections Home Assignments Comments and additional links 0a Mathematical background: Sets, Counting, Series and Summations. - - 0b Mathematical background: Basic integration review, Complex Numbers, Fourier Transforms. - 0c Mathematical background: Convolutions, linear time invariant systems, frequency response. - 1a Introduction: Stochastic Modelling – an uncertain world – scalar uncertain results. 1.1 1.2 1.3 - 1b Introduction: Stochastic Modelling – an uncertain world – random signals, random packets and other models. 1.4 1.5 2a Probability spaces: Generating Uniforms (simulation), Events, probability function, basic axioms. 2.7 2.1 2.2 HW1partialSolution.pdf 2b Probability spaces: Independence, conditional probability, law of total probability, Bayes’ rule. 2.4 2.5 3a Random Variables: Discrete random variables, PMFs, CDFs, expectations: mean, moments, variance, Generation (simulation). 3.1 3.2 4.1 3.3 3b Random Variables: Continuous random variables, PDFs (density), expectations: mean, moments, variance, Markov inequality. 4.1 4.2 4.3 4.6 3c Random Variables: functions of random variables, inverse probability transform, Random number generation, solving basic problems by basic simulation. 4.5 4.9 3.6 4a Useful (families) of discrete distributions: Discrete uniform, binomial, geometric, hyper-geometric, negative binomial, Poisson. 3.5 4b Useful (families) of continuous distributions: Uniform, exponential, normal (Gaussian), gamma, beta, Cauchy. 4.4 Test on Sections 1-4 5a Joint distributions: Joint distribution of pairs (discrete and continuous). Random vectors. Independence. Joint moments. Joint Gaussian. 5.1 6.1 5.2 5.4 5.5 5.9 5b Joint distributions: Conditional distributions, expectation and variance. Conditional distribution of Gaussians. Mean vector and covariance matrix. Jointly Gaussian random vectors. 5.7 5.9 6.3.1 6.4 5c Joint distributions: Transformations and generation. 5.8 5.10 6.2 6.32 6.41 6a Estimation: MAP and ML Estimators, Minimum MSE Linear Estimators and Minimum MSE Estimators. 6.5 6b Estimation: Estimation using a vector observations, examples, parameter estimation, ML parameter estimation, Cramer-Rao (in brief). 6.5 8.2 8.3 7a Transforms, Sums and long term averages: Transforms of random variables, distribution of i.i.d sums, using the DFT to calculate the distribution. 4.7 7.1 7.6 7b Transforms, Sums and long term averages: Law of large numbers, central limit theorem, renewal counting processes. 7.2 7.3 7.5 8a Random process basics: Definitions, distribution, mean function, covariance function, multiple processes, simple examples (Binomial process etc..). 9.1 9.2 9.3 8b Random processes basics: Poisson and related processes, Gaussian processes, Brownian motion (in brief), generating random processes. 9.4 9.5 9.10 9a Signal Level Processes: Stationary processes (and wide sense stationary), cyclostationary processes, time averages and ergodic theorems (non-rigorous). 9.6 9.8 9b Signal Level Processes: Fourier series and Karhunen-Loeve expansions. 9.9 6.3.4 9.9.1 9.10.2 10a Signal Processing Basics: Power spectral density, response of linear systems to signals, sampling and modulation (in brief). 10.1 10.2 10.3 10b Signal Processing Basics: Optimum linear systems, the Kalman filter. 10.4 10.5 11a Cancelled Packet Level Processes: Idea of Markov processes, discrete time Markov chains, steady state, brief description of classes and recurrence properties. 11.1 11.2 11.3 HW Project #6 Markov Chains and Queues cancelled 11b Cancelled Packet Level Processes: Continuous time Markov chains, birth-death chains with queueing examples, numerical techniques (steady state solutions and simulation). 11.4 11.6 12.3 12.4 12a Cancelled Queuing Theory: Terminology, Little’s formula, M/M/1 FCFS waiting time distribution. 12.1 12.2 12.32 12b cancelled Queuing Theory:  M/G/1 queue (in brief), Jackson Networks (in brief), simulation and data analysis of queuing systems. 12.6 12.9 12.10