- 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