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###### Electronics & Communication Engineering Probability and Random Processes
 List Lectures
 # Lecture Name 1 Introduction to the Theory of Probability 2 Axioms of Probability 3 Axioms of Probability (Contd.) 4 Introduction to Random Variables 5 Probability Distributions and Density Functions 6 Conditional Distribution and Density Functions 7 Function of a Random Variable 8 Function of a Random Variable (Contd.) 9 Mean and Variance of a Random Variable 10 Moments

 Title: Probability and Random Processes Department: Electronics & Communication Engineering Author: Prof. Mrityunjoy Chakraborty University: IIT Kharagpur Type: WebLink Abstract: 1. Introduction to Probability  Definitions, scope and history; limitation of classical and relative'frequency'based definitions  Sets, fields, sample space and events; axiomatic definition of probability  Combinatorics: Probability on finite sample spaces  Joint and conditional probabilities, independence, total probability; Bayes rule and applications 2. Random variables  Definition of random variables, continuous and discrete random variables, cumulative distribution function (cdf) for discrete and continuous random variables; probability mass function (pmf); probability density functions (pdf) and properties  Jointly distributed random variables, conditional and joint density and distribution functions, independence; Bayes rule for continuous and mixed random variables  Function of random a variable, pdf of the function of a random variable; Function of two random variables; Sum of two independent random variables  Expectation: mean, variance and moments of a random variable  Joint moments, conditional expectation; covariance and correlation; independent, uncorrelated and orthogonal random variables  Random vector: mean vector, covariance matrix and properties  Some special distributions: Uniform, Gaussian and Rayleigh distributions; Binomial, and Poisson distributions; Multivariate Gaussian distribution  Vector'space representation of random variables, linear independence, inner product, Schwarz Inequality  Elements of estimation theory: linear minimum mean'square error and orthogonality principle in estimation;  Moment'generating and characteristic functions and their applications  Bounds and approximations: Chebysev inequality and Chernoff Bound 3. Sequence of random variables and convergence:  Almost sure (a.s.) convergence and strong law of large numbers; convergence in mean square sense with examples from parameter estimation; convergence in probability with examples; convergence in distribution  Central limit theorem and its significance 4. Random process  Random process: realizations, sample paths, discrete and continuous time processes, examples  Probabilistic structure of a random process; mean, autocorrelation and autocovariance functions  Stationarity: strict'sense stationary (SSS) and wide'sense stationary (WSS) processes  Autocorrelation function of a real WSS process and its properties, cross'correlation function  Ergodicity and its importance  Spectral representation of a real WSS process: power spectral density, properties of power spectral density ; cross'power spectral density and properties; auto'correlation function and power spectral density of a WSS random sequence  Linear time'invariant system with a WSS process as an input: sationarity of the output, auto'correlation and power'spectral density of the output; examples with white'noise as input; linear shift'invariant discrete'time system with a WSS sequence as input  Spectral factorization theorem  Examples of random processes: white noise process and white noise sequence; Gaussian process; Poisson process, Markov Process

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