CS723 : Probability and Stochastic Processes

Course Overview

Course Synopsis

This is a graduate level course. The course will start by presenting fundamental concepts of probability theory. It will then develop mathematically sound concepts of random variables and their processing through PDF and CDF.

Course Learning Outcomes

Upon successful completion of this course, students should be able to:

  • Feel comfortable about concepts and terminology of probability theory and its domains of application.
  • Apply set-theoretic probabilistic modeling of un-predictable phenomena and academic and real-life problems.
  • Solve simple problems related to random variables, their distribution functions, expected values, moments, and their conditional expectations.
  • Work with jointly distributed pairs of random variables using their joint and marginal densities.
  • Understand how sequence of random variables behave and converge to predictable behaviour.

Course Calendar

Introduction to Probability and Stochastic Processes1Papoulis (2002), Krishnan (2006)
Set Theory2do.
Relations, Functions, Probability3do.
Probability Experiments: Repeated, Dependent, Cascaded4do.
Joint Probability, Conditional Probability, Total Probability5do.
Combinatronics, Birthday Problem, Bayes Rule6do.
Bayes Rule Application, Partitioned Footpath, Partioned Floor, Trains on a Junction7do.
Trains on a Junction, A local bus stop, Random line partition8do.
Assignment No. 1
Equally likely outcomes, Random Variables, Chuk-a-luck, Binomial & Poisson Distributions9do.
CDF of Poisson PMF, Joint Distribution Example, Marginal PMF10do.
Conditional Distribution, Conditional PMF, Expected Value, Transformation of Random Variables11do.
Transformation of RVs, Conditional Expectation, Co-variance and Correlation12do.
Continuous Random Variable, Probability Density Function, PDF and CDF of a Continuous RV13do.
CDF of a Continuous RV, CDF and PDF of Nozzle Height, Exponential RV14do.
Failure of TV Set Example, Gaussian RV15do.
Mean of Transformed RV, Height Distribution of Humans Example, Higher Moments16do.
Assignment No. 2
Scaling of Gaussian RV, Standard Gaussian Events, Joint Distributions, Pair of RVs17do.
Pair of RVs , Buffon's Needle, Exponential RV Pair, Pair of Erlang RVs18do.
Exponential RV Pair, Pair of Dependent RVs, Correlation & Covariance19do.
Correlation & Covariance, Correlated Gaussian RVs20do.
Function of two RVs, Evaluation of Fz(Z), CDF & PDF of Z = X + Y21do.
Function of two RVs, Evaluation of Fz(Z), CDF & PDF of Z = Y / X, CDF & PDF of Z = max (X, Y)22do.
Mid-term Examination
Multiple derived RVs, Probability Computation23do.
Pair of derived RVs24do.
Direct Computation of Derived CDF25do.
Non-invertible Transformations26do.
Moments of U and V from f(x, y)27do.
Expectation of Transformed Random Variables, Covariance Matrix, Eigenvalues & Eigenvectors28do.
Vector RVs, Transforming Vector RVs29do.
PDF of Z = X+X ? 2X, Characteristic Functions30do.
Convergence of Sequence, Convergence of RVs31do.
Norm of Vectors and Function, Convergence of RVs, Inequalities32do.
Basics of Markov chains, Examples of Markov chains33Lawler (2006)
Transition probabilities, Stochastic matrix, Stationary probability distribution34do.
Periodic and aperiodic Markov chains, Multivariate joint probability35do.
Transient and recurrent Markov chains, Return times of recurrent states36do.
Course Viva
n-step stochastic matrix, n-step reachability37do.
Diagonalization of stochastic matrix, Periodic and aperiodic recurrence38do.
Academic Research Paper
Eigen-analysis of stochastic matrix39do.
Bipartite and tripartite graphs of Markov chains40do.
Asymptotic behaviour of Markov chains41do.
Partitioning of Markov chains42do.
Markov chains with infinite state space43do.
Random walk on real line, Random walk on 2-d plane, Random walk in higher dimensions, Continuous-time Markov chains44do.
Continuous-time Markov chains, Poisson arrival and departure processes, Stationary probability distribution, Queuing theory45do.
Final-term Examination
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