**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