Abstract: 
1. Introduction to Probability
? Definitions, scope and history; limitation of classical and relativefrequencybased
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
? Vectorspace representation of random variables, linear independence, inner product, Schwarz Inequality
? Elements of estimation theory: linear minimum meansquare error and orthogonality principle in estimation;
? Momentgenerating 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: strictsense stationary (SSS) and widesense stationary (WSS) processes
? Autocorrelation function of a real WSS process and its properties, crosscorrelation
function
? Ergodicity and its importance
? Spectral representation of a real WSS process: power spectral density, properties of power spectral density ; crosspower spectral density and properties; autocorrelation function and power spectral density of a WSS random sequence
? Linear timeinvariant system with a WSS process as an input: sationarity of the output, autocorrelation and powerspectral density of the output; examples with whitenoise as input; linear shiftinvariant discretetime 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
