Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
ESSENTIALS OF PROBABILTY
MATH-GA.2901-001
Description: An introduction to the mathematical treatment of random phenomena oc-
curring in the natural, physical, and social sciences. Topics will include probability spaces,
random variables, distributions, law of large numbers, central limit theorem, random walk,
Markov chains and martingales in discrete time, and if time allows diffusion processes in-
cluding Brownian motion.
Prerequisites: Calculus through partial derivatives and multiple integrals; no previous
knowledge of probability is required.
Textbooks: We will closely follow Probability Essentials, by J. Jacod and P. Protter
(Springer). The book is available to you digitally.
Course requirements and grading: The course requirements are
• Homework: There will be one assignment every week. Each assignment will count
equally toward your grade (regardless of the max possible points). Taken together,
the homework will be worth 40% of your grade. Late homework will be deducted 10%
per day, unless a request for an extension has been granted.
• Exams: The midterm exam will be worth 20% of your overall grade, and the final
exam will be worth 40% of your overall grade.
Some Policies
Collaboration on homework: Students are welcome – and even encouraged – to dis-
cuss the homework problems with others. However, each student must implement
and present his or her own solutions (this is an important part of the learning pro-
cess). Direct copying of another student’s solution is not permitted – both because
it amounts to cheating, and because it defeats the entire purpose of the homework
(which is to gain familiarity with new concepts and techniques).
Academic integrity: Plagiarism and cheating will not be tolerated. NYU’s College of
Arts and Sciences has policies in this area, and they will be followed. See
http://cas.nyu.edu/academic-integrity.html
1
Tentative Topic List
The class will follow closely the book Probability Essentials, by J. Jacod and P. Protter
(Springer). Topics to be covered include :
1. Axioms of Probability
2. Conditional Probability and Independence
3. Probabilities on a Finite or Countable SpaceSpace
4. Random Variables on a Countable Space
5. Construction of a Probability Measure
6. Construction of a Probability Measure on R
7. Random Variables
8. Integration with Respect to a Probability Measure
9. Independent Random Variables
10. Probability Distributions on R
11. Probability Distributions on mathbbRn
12. Characteristic Functions
13. Properties of Characteristic Functions
14. Sums of Independent Random Variables
15. Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)
16. Convergence of Random Variables
17. Weak Convergence
18. Weak Convergence and Characteristic Function
19. The Laws of Large Numbers
20. The Central Limit Theorem
21. L2 and Hilbert Spaces
22. Conditional Expectation
23. Martingale
24. Supermartingales and Submartingales
25. Martingale Inequalities
26. Martingale Convergence Theorems
27. The Radon-Nikodym Theorem
28. Markov chains