Module 1: Introduction to probability: Random experiment, sample space, events and outcome, probability models, axioms and properties of probability, joint probability, conditional probability, total probability, Bayes' theorem, independent events, combined experiments, Bernoulli trials, De-Moivre-Laplace approximation, Poisson approximation.
(3 hours)

Module 2: Introduction to random variables: Continuous and discrete random variables, definitions and properties of cumulative distribution function (cdf) and probability density function (pdf), probability mass function (pmf), cdfs and pdfs of some standard continuous/discrete random variables, properties of conditional cdf and pdf, law of large numbers, central limit theorem and its significance.
(5 hours)

Module 3: Operations on random variables: Operations on one random variable: Expectation, conditional expectation, moments, variance, skewness, Chebyshev's and Markov's inequalities, characteristic function and moment generating function, moments theorem, Chernoff's inequality and bound, cumulants, transformation for function of one random variable, Random vectors: bivariate and multivariate random variables, properties of joint and conditional cdfs and pdfs, Operations on multiple random variables: Expectation, joint moments, calculation of correlation and covariance, marginal cdfs and pdfs, joint and marginal characteristic functions, moments theorem, Functions of random variables: transformation for one function of two random variables, transformation for two functions of two random variables, calculation of joint and marginal cdfs and pdfs for transformed random variables.
(12 hours)

Module 4: Random processes: classification of random processes, first-order stationary process, second-order stationary process, wide-sense stationary (WSS) process, N-order and strict sense stationary processes, statistical characterization of random process, WSS process though LTI system, matched filter, Wiener filter, mean-ergodic and correlation-ergodic processes, autocorrelation and cross-correlation, auto-covariance and cross-covariance, power density spectrum and cross-power density spectrum, Poisson process, Renewal process, Wiener Process.
(6 hours)

Module 5: Introduction to discrete time Markov chain: recurrent and transient states, limiting n-step transition probabilities Convergence: convergence in mean of order p, Karhunen Love expansion, Wiener integrals, orthogonality principle, projection theorem, spectral representation.
(4 hours)

To develop understanding of fundamentals of probability theory and its applicability to real world engineering problems.

To develop understanding about random experiments, random variables, statistical characterization of random quantities.

To develop understanding about various statistical operations on random variables such as mean, variance, skewness, correlation, etc. applicable to engineering problems.

To develop understanding about different types of random processes and their temporal and spectral characteristics for modelling and analysis of physical problems in the field of communication theory, signal processing, computer networks, etc.

After the completion of this course, students will be able to:

CO1: know the key concepts of probability theory, discrete and continuous random variables, distribution function, and density functions.

CO2: define and apply standard statistical distributions i.e., Normal, Binomial, Poisson, etc. to model various physical phenomena described by them.

CO3: acquire analytical skills for performing basic operations on random variables such as conditioning, independence, expectation, moments, characteristic functions.

CO4: perform transformations on univariate, bivariate, and multivariate random variables.

CO5: get knowledge of types of random processes, stationarity concepts, and temporal and spectral characteristics, response of LTI systems to random inputs.

Peyton Z. Peebles, Jr., Probability, random variables and random signal principles, McGraw Hill , 4th Edition 2001

John A. Gubner, Probability and random processes for electrical and computer engineers, Cambridge university press , 1st Edition 2006

A. Papoulis and S. U. Pillai, Probability random variables and stochastic processes, McGraw Hill , 4th Edition 2017

Sheldon M. Ross, Introduction to probability and statistics for engineers and scientists, Academic Press , 5th Edition 2014