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Math 36a - Probability - Summer 2024
Attendance: Attendance at all classes is expected, even if recordings are made available.
Course description and objective: Recent event just confirmed that, in nature, some phenomena are unpredictable. In fact, rigorously, most phenomena cannot be said for sure to occur or not. Think about it for a minute: which natural phenomenon can you name that is fully predictable? In general, we can rather estimate the chance, or, in mathematical terms, the probability, for an outcome to occur. Appropriately handling uncertainty is thus essential in a variety of domains including economics, investing strategies on the stock market, strategies to beat the casino, machine learning and artificial intelligence, game theory, political polls, and, in fact, any medical or societal statistics one is exposed everyday in newspapers and internet. Probability theory is the axiomatic mathematical formalization of these uncertain events.
This course introduces the foundations of mathematical probability, completed with plenty of real world examples. The main topics of study will be: combinatorics (how to count things), random variables (‘events’ with chances of occurring), conditional probability (‘what are the chancesthat it is currently raining given that I see my friend taking their umbrella?’), discrete and continuous distributions, jointly distributed random variables, moments of random variables and thebasic limit theorems (strong/weak law, central limit theorem).
The material presented in this class is original and does not follow a specific textbook. Lecture notes will be provided at the end of each chapter. No textbook is required.
Interesting references include:
- Elementary Probability Theory, by KL Chung and F. AitSahlia (Springer) [closest to class level]
- An Introduction to Probability Theory and its Applications, Vol. 1&2, William Feller (Willey) [more theoretical than the class]
- First Course in Probability by Sheldon Ross (Pearson) [least theoretical]
Learning goals: Upon successful completion of Math 36A students will be able to:
- Solve basic counting problems
- Compute probabilities on a set theoretic basis & use the axioms of probability
- Compute conditional probabilities and check independence of events
- Define random variables for simple random experiments
- Calculate probabilities, expectations and variances of discrete and continuous random variables
- Perform simple computations involving jointly distributed random variables
- Use the weak and strong laws, and the central limit theorem to approximate probabilities
- Set theory, combinatorics, probability: 6/3 - 6/6
- Discrete Random Variables and Conditional Probability: 6/10 - 6/18
- Take Home Midterm: 6/24. Expected duration: 90 minutes.
- Conditional Probability, Independence: 6/19 - 6/25
- Continuous Random variables, Limit Theorems: 6/26-7/3
- Review session: 7/4
- Exam - set by the registrar, either 7/6 or 7/7
- The grade will be based at 30% on the weekly homework assignments, 30% on a midterm exam and 40% on a final examination.
- Homework and midterm will be submitted on Gradescope. You should have been already registered. If you have not been notified of your registration to gradescope, please email me asap.
- Bonus up to 5 points will be granted based on significant participation. Participation includes answering instructor’s questions in class, asking questions in class, posting to asynchronous discussions online on the piazza forum, speaking during in-class discussions, raising questions as soon as ideas become unclear, responding thoughtfully and constructively to the contributions made by other class members, reading and responding to course-related e-mails and posts.
- The lowest homework grade will be dropped.
- Collaboration and discussion on homework is encouraged but you must write up your solutions independently of your classmates.
- Collaboration on exams is forbidden and will be strictly enforced; any suspicion will be reported to academic integrity (see below).
Accommodations for Students with Disabilities: I am more than happy to support any ac commodation students may need, and it is my objective to make this class meaningful for everyone. If you are a student who needs accommodations as outlined in a Brandeis accommodation letter, please talk with me as early as possible and present your letter of accommodation as soon as you can, as I can provide accommodation in advance, but not retroactively. If you have questions about documenting a disability or requesting accommodations, please contact Student Accessibility Support (SAS) at 781.736.3470 or [email protected].