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Electronics & Communication Engineering  Probability and Random Processes
Title: Probability and Random Processes
Department: Electronics & Communication Engineering
Author: Prof. Mrityunjoy Chakraborty
University: IIT Kharagpur
Type: video

1. Introduction to Probability

? Definitions, scope and history; limitation of classical and relative-frequency-based

? 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

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

? 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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