Math 36a - Probability - Summer 2024

Course description and objective: Recent event just confirmed that, in nature, some phenom- ena 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 in-cluding 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 theory was initiated by two French mathematicians, B. Pascal and P. de Fermat, as they were corresponding about games of chance; these problems and seminal works continued to in且uence outstanding scientists including Huygens, Bernoulli, and DeMoivre, leading to establishing the bases of the modern mathematical theory of probability. Today, probability theory is a well established branch of mathematics.  It is an active area of fundamental research, and, of course, of applied mathematics development dealing with economics, finance, neurobiology, physics, ecology, climate change, medical treatments or pandemics.

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), ran- dom variables (‘events’ with chances of occurring), conditional probability (‘what are the chances that it is currently raining given that I see my friend taking their umbrella?’), discrete and con- tinuous distributions, jointly distributed random variables, moments of random variables and the basic limit theorems (strong/weak law, central limit theorem).

Textbooks (recommended)

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]

Expectation of students’ efort:  Success in this course is based on the expectation that students will spend a minimum of 9 hours of study time per week in preparation for the class (reviewing class material, completing homework, preparation for exams, etc.).

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 vari- ables

•  Perform simple computations involving jointly distributed random variables

•  Use the weak and strong laws, and the central limit theorem to approximate probabilities

Course Plan

•  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

Grading Policy:

•  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 contri- butions 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 writeup 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).