Prerequisite course : Familiarity with basic probability

Learning Objective

Modeling uncertainty is essential for effective decision making in many real life applications. Probability models provide the necessary framework to model uncertainty. The course aims to provide an exposure to basic concepts of probabilistic models and how it can be used to solve real problems.

Learning Outcomes

On successful completion of the course, the student will be able to

  • state important definitions, and prove results in probability theory
  • write codes for simulation studies
  • formulate probabilistic models given real problems
  • develop randomized algorithms

Course Content

Probability spaces, Axioms of Probability, Continuity of probability, Random variables, Common distributions, Distribution functions, Multiple random variables and Joint distributions, Functions of random variables, Moments, Conditional probability/expectation, Bayes rule, Sequences of random variables and convergence concepts, Laws of large numbers, Central limit theorem (3 weeks)

Stochastic process: Poisson process, Hazard functions, Random walk and Markov chain (2 weeks)

Estimators: maximum likelihood, maximum a posteriori ​Concentration inequalities:​ Markov, Chebyshev, Hoeffding, Chernoff inequalities (2 weeks)

Simulation: Inverse transformation method, Techniques for simulating continuous random variables, MCMC importance sampling, Rejection sampling, Gibbs sampling (3 weeks)

Models: Linear regression, hidden Markov models (2 weeks)

Randomized algorithms: Monte Carlo and Las Vegas methods, randomized quick sort, randomized algorithm for satisfiability (SAT). (2 weeks)

Text books

  1. Ross, Sheldon M. ​ Introduction to probability models. ​ Academic press, 2014. ​ ISBN-13​ :​9780124079489

  2. Ross, Sheldon M. ​A First Course in Probability. Pearson, Ninth edition, 2014. ISBN: 9789353065607.

References

  1. Feller, Willliam. An Introduction to Probability Theory and its Applications. Vol.1. ​Wiley;​ ​3rd Edition, 2008. I​SBN-13: 9780471257080

  2. P.G.Hoel, S.C.Port and C.J.Stone, ​ Introduction to Probability Theory. ​Houghtion Mifflin; 1 edition​ ​1971. ISBN-13: 9780395046364

  3. Bertsekas, Dimitri P., and John N. Tsitsiklis. ​ Introduction to Probability. Vol. 1. Athena Scientific, 2002. ISBN-13: 9781886529236

  4. Rohatgi, Vijay K., and AK Md Ehsanes Saleh. ​ An Introduction to Probability and Statistics. John Wiley & Sons, 2015.​ ​ ISBN-13: 9781118799642

Past Offerings

  • Offered in July-Dec, 2019 by Mrinal