Probability Foundations for Electrical Engineers

July-November 2017

I am releasing the PDF notes that are meant to accompany the NPTEL video course. I am grateful to the scribes and TAs who worked for many hours typing up these class notes – most of them are acknowledged by name in the PDF files.

I would like to inform the reader that these notes have not been carefully edited, so you may find the occasional bug, typo, inadequate citation, or less than perfect language in some places. I would be grateful if such shortcomings are brought to my notice.

Module 0: Preliminaries

1. Basic Set Theory

2. Basic Real Analysis

3. Cardinality and Countability

Module 1: Probability Measures

4. Probability Spaces

5. Properties of Probability Measure

6. Discrete Probability Spaces

7. Borel Sets and Lebesgue Measure

8. The Infinite Coin Toss Model

9. Conditional Probability and Independence

10. The Borel-Cantelli Lemmas

Module 2: Random Variables

11. (i) Random Variables (ii) Types of Random Variables

12. Multiple Random Variables and Independence

13. Conditional Distributions, Joint Continuity

14. Introduction to Transformation of Random Variables

15. Sums of Random Variables

16. General Transformations

Module 3: Integration and Expectation

17. Integration and Expectation

18. Properties of Integrals

19. The Monotone Convergence Theorem

20. Expectation of Discrete RVs, Expectations over Different Spaces

21. Expectation of Continuous RVs, Fatou’s Lemma, Dominated Convergence Theorem

22. Variance and Covariance

23. Conditional Expectation

Module 4: Transforms

24. Probability Generating Function

25. Moment Generating Function

26. Characteristic Function

27. Concentration Inequalities

Module 5: Limit Theorems

28. Convergence of Random Variables

29. The Law of Large Numbers

30. The Central Limit Theorem