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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: video 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