Course: Probability Theory (KSMIC04)
Timings: Mondays and Saturdays 9:00 am to 11:00 am.
Teaching Assistant: Arunkumar
Course meeting venue: Google meet link
Syllabus: Probability space, events. Axioms of probability. Inclusion exclusion principle. Combinatorial examples.
Independence and conditioning. Bayes formula. Random variables. Distribution functions.
Examples: Binomial, Geometric, Poisson, Hypergeometric etc. Expectation, variance and covariance, generating functions and characteristic function.
Independence and conditioning of random variables. Joint disribution, Distribution of the sum.
Continous distributions and densities. Examples: Normal, exponential and gamma, uniform and beta, etc.
Inequalities: Markov, Chebyshev, Cauchy-Schwarz, Bonferroni. IID random variables. Weak law of large numbers, CLT. Simple symmetric random walk, reflection priciple, recuurence and transience.
Grading: Biweekly quiz (20%), Mid-term test (30%), End-term test (50%)
References and Resources:
- Manjunath Krishnapur’s Lecture Notes
- William Feller An introduction to probability theory and its applications - Vol. 1, 2
- Sheldon Ross A first course in probability
- Seeing Theory: A visual introduction to probability and statistics Has several interactive illustrations.
- Arnab Chakraborty’s Notes on Random Walks
Classroom Scribes
| Lecture | Date | Contents |
|---|---|---|
| 1 | 1 November 2021 | Introduction, Examples, Probability Spaces |
| 2 | 6 November 2021 | Axioms of probability and illustrations |
| 3 | 8 November 2021 | Inclusion-exclusion principle and examples |
| 4 | 13 November 2021 | Bonferroni inequalities |
| 5 | 15 November 2021 | Independence and conditional probability |
| 6 | 20 November 2021 | Bayes’ rule and applications |
| 7 | 22 November 2021 | Paradoxes, resolutions and other issues |
| 8 | 27 November 2021 | Discrete probability distributions and examples |
| 9 | 29 November 2021 | Continuos probability distributions and examples |
| 10 | 4 December 2021 | Joint distributions and change of variable formula |
| 11 | 6 December 2021 | Expectation, moments and generating functions |
| 12 | 11 December 2021 | Inequalities (Cauchy-Shwarz, Markov, Chebyshev, Chernoff ) |
| 13 December 2021 | Mid-Semester | |
| 18 December 2021 | Mid-Semester | |
| 13 | 20 December 2021 | Conditional distributions and conditional expectation |
| 14 | 23 December 2021 | Make up lecture |
| 15 | 27 December 2021 | Characteristic function and properties |
| 16 | 1 January 2022 | Law of large numbers |
| 17 | 3 January 2022 | Gaussian random variables and properties |
| 18 | 8 January 2022 | Poisson limits |
| 19 | 10 January 2022 | Central limit theorem |
| 20 | 15 January 2022 | Central limit theorem (contd.) |
| 21 | 17 January 2022 | Simulation |
| 22 | 22 January 2022 | Simple symmetric random walk |
| 23 | 24 January 2022 | Recurrence and transience |
| 24 | 29 January 2022 | Reflection Principle |
| 12 February 2022 | End Semester |
| Problem Sets: | Set-1 | Set-2 | Set-3 | Set-4 | Set-5 | Set-6 | Set-7 | Set-8 |