Course Description
The course is broadly divided into 2 modules:
- Module 1 Random Number Generation
- Module 2 Random Processes
Course Syllabus
Review of basic probability: Random variables and random vectors, Classical Inequalities and limit theorems
Random Number Generation; Generation of Random Variables: Inverse Transform method, Acceptance- rejection method, Variance Reduction methods: Control Variate, Conditioning, Importance Sampling; Uncertainty, Entropy.
Random Processes: Definition and classification of random processes, Autocorrelation and properties, Random process through LTI systems, Bernoulli processes, Markov Chains (MCs): Preliminaries, Discrete-time MC: Transition Probability Matrix, Classification of states, Chapman-Kolmogorov Equation, Limiting & stationary Distributions, Ergodic MC; Continuous time MC: Poisson Process, Weiner process, Birth and Death Processes; Application and Case Studies.
Course Logistics
- Schedule: Slot E, 8:00 am - 8:55 am Tuesday, 9:00 am - 9:55 am Wednesday, 10:00 am - 10:55 am Thursday
- Venue: 5205, Core 5.
Course Evaluation
- Quizzes: 30%
- Mid semester exam: 30%
- End semester exam: 30%
- Participation: 10%
Some references (not an exhaustive list)
- Ross, S.M., 2022. Simulation. Academic Press.
- Ross, S.M., 1995. Stochastic processes. John Wiley & Sons.
- Bertsekas, D. and Tsitsiklis, J.N., 2008. Introduction to probability (Vol. 1). Athena Scientific.
- Blitzstein, J.K., and Hwang J., 2019. Introduction to probability. Taylor & Francis Group, LLC.
Tentative Lecture Plan
| Lecture | Date | Topic |
|---|---|---|
| Lecture 1 | 6-Jan-2025 | Introduction to Monte Carlo Simulations |
| Lecture 2 | 7-Jan-2025 | Pseudorandom number generators |
| Lecture 3 | 8-Jan-2025 | Inverse transform for discrete random variables |
| Lecture 4 | 13-Jan-2025 | Accept-Reject sampling for discrete random variables |
| Lecture 5 | 15-Jan-2025 | Composition method |
| Lecture 6 | 20-Jan-2025 | Inverse transform for continuous random variables |
| Lecture 7 | 21-Jan-2025 | Accept-Reject sampling for continuous random variables |
| Lecture 8 | 22-Jan-2025 | Accept-Reject sampling for continuous random variables |
| Lecture 9 | 27-Jan-2025 | Quiz 1; Accept-Reject sampling for continuous random variables |
| Lecture 10 | 28-Jan-2025 | Box Muller and Ratio of Uniforms |
| Lecture 11 | 3-Feb-2025 | Ratio of Uniforms |
| Lecture 12 | 4-Feb-2025 | Miscellaneous methods in sampling |
| Lecture 13 | 5-Feb-2025 | Variance reduction techniques |
| Lecture 14 | 10-Feb-2025 | Variance reduction techniques |
| Lecture 15 | 11-Feb-2025 | Simple Monte Carlo, Importance sampling |
| Lecture 16 | 12-Feb-2025 | Importance sampling |
| Lecture 17 | 17-Feb-2025 | Optimal Importance sampling |
| Lecture 18 | 18-Feb-2025 | Quiz 2; Uncertainty; Entropy |
| Lecture 19 | 19-Feb-2025 | |
| Lecture 20 | 25-Feb-2025 | Introduction to random processes |
| Lecture 21 | 26-Feb-2025 | Bernoulli Process |
| 6-Mar-2025 | Mid Semester Examination |