Please find an updated version of the lecture notes. Have nice holidays and a good preparation for the exam.
In order to help you preparing for the final exam, we have created a mock exam. Click on this text to download it. The mock exam follows the same format and typology of questions as the final exam.
Information about the course
This is the first announcement about the plan for the course Probability II. The goal is to study (discrete) stochastic process, that is a sequence of random variables defined on a common probability space. For concreteness, I will focus on real-valued random variables and I plan to discuss several applications during the lectures and exercise sessions.
There will be three main topics:
1- Conditional expectation
2- Theory of martingales
3- Markov chains
These concepts are fundamental both in theoretical and applied probability. In particular, they have crucial applications in mathematical finance, data science and mathematical physics. From a mathematical viewpoint, this course relies on analysis, especially measure theory, but also has connections to linear algebra and combinatorics.
In the first week, I will begin this course by a review of Probability I. In the first lecture, this includes a short introduction to the foundations of probability theory, such as the key concepts of Lebesgue measure, random variables, probability law, independence, etc. I also plan to review the main modes of convergence for sequences of random variables. In the second lecture, I will review the main results from Probability I: the Borel-Cantelli Lemma, the law of large numbers and central limit Theorems. I also plan to recall the concept of characteristic functions.
Information about the exercise classes
Exercise classes (and lectures) will be streamed live and recorded.
Submission of weekly assignment sheets is optional.
There is no required minimum number of points to access the final exam.
To access the file "Markov chain simulation" first you need to download the file. Your browser may mistaken the file for text and open it in a new browser window. In order to avoid this and in order to actually download the file, you must right-click on the link and select "Save linked content as" or "Save link as" or a similar item (the actual name depends on your browser and language), then download it in a folder in your computer.
Go to the website https://jupyter.org/try and click on "Try JupyterLab". A demo notebook will open in your browser (give it some time to load). Upload the Notebook you just downloaded and open it.
For a more comfortable experience, please consider downloading Jupyter from their website and install the software on your computer.
Notice that if you installed Python through Anaconda, Jupyter is already installed on your machine.
In case you have problems with Jupyter (either online or installed), you can download the pdf copy of the notebook above, but you will lose the interactivity of the code.
I just added a first draft of the notes for the course on my webpage: http://user.math.uzh.ch/gaultier/
These notes include the review from Probability 1 and the first part of the course on conditional expectation.
Please let me know if you find some typos or inconsistencies in the notes. Thanks in advance!
There is an ongoing student course evaluation conducted by UZH that you are encourage to answer. Here is a link to this https://idevasys03.uzh.ch/evasys_02/public/online/index/input?p=HFV14&nLangID=1852
I just added a new version of the lecture notes which includes the whole section of the course on martingale theory. It is also available on my webpage:http://user.math.uzh.ch/gaultier/
I have just added an updating version of the lecture notes. Please let me know in case you find typos.
Modul: 08.02.2021 9:00-12:00, Raum: Mehrere Räume Plätze: ?, Typ: schriftlich
Y27 H12, H25
Repetition: 05.08.2021 9:00-12:00, Raum: Y27H12 Plätze: 50, Typ: schriftlich
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