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Prob-Class-Notes

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Probability-The Science_of_Uncertainty_and_Data taught by the Institute for Data, Systems, and Society (IDSS) MIT faculty Professor John Tsitsiklis

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# MITx-6.431x_Probability-The-Science_of_Uncertainty_and_Data
Probability-The Science_of_Uncertainty_and_Data taught by the Institute for Data, Systems, and Society (IDSS) MIT faculty Professor John Tsitsiklis

[Credential Link](https://credentials.edx.org/records/programs/shared/dd1f16009d8e4e54a762ce4a07020f45/)

![](https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/master/MIT%20certificate.PNG)

# 1. Introduction and Course Team

### Welcome to 6.431x, an introduction to probabilistic models, including random processes and the basic elements of statistical inference.

The world is full of uncertainty: accidents, storms, unruly financial markets, noisy communications. The world is also full of data. Probabilistic modeling and the related field of statistical inference are the keys to analyzing data and making scientifically sound predictions.

The course covers all of the basic probability concepts, including:

* multiple discrete or continuous random variables, expectations, and conditional distributions

* laws of large numbers

* the main tools of Bayesian inference methods

* an introduction to random processes (Poisson processes and Markov chains)

## 2. Course objectives
Upon successful completion of this course, you will:

### At a conceptual level:

* Master the basic concepts associated with probability models .

* Be able to translate models described in words to mathematical ones.

* Understand the main concepts and assumptions underlying Bayesian and classical inference .

* Obtain some familiarity with the range of applications of inference methods .

### At a more technical level:

* Become familiar with basic and common probability distributions .

* Learn how to use conditioning to simplify the analysis of complicated models.

* Have facility manipulating probability mass functions , densities , and expectations .

* Develop a solid understanding of the concept of conditional expectation and its role in inference.

* Understand the power of laws of large numbers and be able to use them when appropriate.

* Become familiar with the basic inference methodologies (for both estimation and hypothesis testing ) and be able to apply them.

* Acquire a good understanding of two basic stochastic processes (Bernoulli and Poisson) and their use in modeling.

* Learn how to formulate simple dynamical models as Markov chains and analyze them.

## Professor John Tsitsiklis
Dr. John Tsitsiklis is a Clarence J Lebel Professor in the Department of Electrical Engineering and Computer Science, and the director of the Laboratory for Information and Decision Systems at MIT.

His research interests are in the fields of systems, optimization, control, and operations research. He is a coauthor of Parallel and Distributed Computation: Numerical Methods (1989, with D. Bertsekas), Neuro-Dynamic Programming (1996, with D. Bertsekas), Introduction to Linear Optimization (1997, with D. Bertsimas), and Introduction to Probability (1st ed. 2002, 2nd. ed. 2008, with D. Bertsekas). He is also a coinventor in seven awarded U.S. patents.

He is a member of the National Academy of Engineering, and a Fellow of the IEEE (1999) and of INFORMS (2007). His distinctions include the ACM Sigmetrics Achievement Award (2016), the INFORMS John von Neumann Theory Prize (2018), and the IEEE Control Systems Award (2018). He holds honorary doctorates from the Universite catholique de Louvain, (2008), the Athens University of Economics and Business (2018), and the Harokopio University.

Professor Tsitsiklis has been teaching probability for over 20 years.

## Professor Patrick Jaillet
Patrick Jaillet is Dugald C. Jackson Professor in the Department of Electrical Engineering and Computer Science and a member of the Laboratory for Information and Decision Systems at MIT.

Professor Jaillet's research interests include online optimization and learning; machine learning; and decision making under uncertainty. Professor Jaillet's teaching covers subjects such as machine learning; algorithms; mathematical programming; network science and models; and probability. Dr. Jaillet's consulting activities primarily focus on the development of optimization-based analytic solutions in various industries, including defense, financial, electronic marketplace, and information technology.

Professor Jaillet was a fulbright scholar in 1990 and the recipient of many research and teaching awards. He is a Fellow of the Institute for Operations Research and Management Science Society (INFORMS), a member of the Mathematical Optimization Society (MOS), and a member of the Society for Industrial and Applied Mathematics (SIAM). He is currently an Associate Editor for INFORMS Journal on Optimization, Networks, and Naval Research Logistics, and has been an Associate Editor for Operations Research from 1994 until 2005 and for Transportation Science from 2002 until 2017.

## Professor Dimitri Bertsekas
Dimitri P. Bertsekas is McAfee Professor of Engineering in the Electrical Engineering and Computer Science Department of MIT. In 2019, he was also appointed a full time professor in the department of Computer, Information, and Decision Systems Engineering at Arizona State University, Tempe, while maintaining a research position at MIT.

His research spans several fields, including optimization, control, large-scale computation, and data communication networks, and is closely tied to his teaching and book authoring activities. He has written numerous research papers, and seventeen books and research monographs, several of which are used as textbooks in MIT classes.

Professor Bertsekas was awarded the INFORMS 1997 Prize for Research Excellence in the Interface Between Operations Research and Computer Science for his book "Neuro-Dynamic Programming", the 2000 Greek National Award for Operations Research, the 2001 ACC John R. Ragazzini Education Award, the 2009 INFORMS Expository Writing Award, the 2014 ACC Richard E. Bellman Control Heritage Award for "contributions to the foundations of deterministic and stochastic optimization-based methods in systems and control," the 2014 Khachiyan Prize for Life-Time Accomplishments in Optimization, and the SIAM/MOS 2015 George B. Dantzig Prize. In 2018, he was awarded, jointly with his coauthor John Tsitsiklis, the INFORMS John von Neumann Theory Prize, for the contributions of the research monographs "Parallel and Distributed Computation" and "Neuro-Dynamic Programming". In 2001, he was elected to the United States National Academy of Engineering for "pioneering contributions to fundamental research, practice and education of optimization/control theory, and especially its application to data communication networks."

Prof Bertsekas has been teaching probability for over 15 years.

## 3. Study guide

A guide on how to use the wealth of available material

This class provides you with a great wealth of material, perhaps more than you can fully digest. This “guide" offers some tips about how to use this material.

Start with the overview of a unit, when available. This will help you get an overview of what is to happen next. Similarly, at the end of a unit, watch the unit summary to consolidate your understanding of the “big picture" and of the relation between different concepts.

Watch the lecture videos. You may want to download the slides (clean or annotated) at the beginning of each lecture, especially if you cannot receive high-quality streaming video. Some of the lecture clips proceed at a moderate speed. Whenever you feel comfortable, you may want to speed up the video and run it faster, at 1.5x.

Do the exercises! The exercises that follow most of the lecture clips are a most critical part of this class. Some of the exercises are simple adaptations of you may have just heard. Other exercises will require more thought. Do your best to solve them right after each clip — do not defer this for later – so that you can consolidate your understanding. After your attempt, whether successful or not, do look at the solutions, which you will be able to see as soon as you submit your own answers.

Solved problems and additional materials. In most of the units, we are providing you with many problems that are solved by members of our staff. We provide both video clips and written solutions. Depending on your learning style, you may pick and choose which format to focus on. But in either case, it is important that you get exposed to a large number of problems.

The textbook. If you have access to the textbook, you can find more precise statements of what was discussed in lecture, additional facts, as well as several examples. While the textbook is recommended, the materials provided by this course are self-contained. See the “Textbook information" tab in Unit 0 for more details.

Problem sets. One can really master the subject only by solving problems – a large number of them. Some of the problems will be straightforward applications of what you have learned. A few of them will be more challenging. Do not despair if you cannot solve a problem – no one is expected to do everything perfectly. However, once the problem set solutions are released (which will happen on the due date of the problem set), make sure to go over the solutions to those problems that you could not solve correctly.

Exams. The midterm exams are designed so that in an on-campus version, learners would be given two hours. The final exam is designed so that in an on-campus version, learners would be given three hours. You should not expect to spend much more than this amount of time on them. In this respect, those weeks that have exams (and no problem sets!) will not have higher demands on your time. The level of difficulty of exam questions will be somewhere between the lecture exercises and homework problems.

Time management. The corresponding on-campus class is designed so that students with appropriate prerequisites spend about 12 hours each week on lectures, recitations, readings, and homework. You should expect a comparable effort, or more if you need to catch up on background material. In a typical week, there will be 2 hours of lecture clips, but it might take you 4-5 hours when you add the time spent on exercises. Plan to spend another 3-4 hours watching solved problems and additional materials, and on textbook readings. Finally, expect about 4 hours spent on the weekly problem sets.

Additional practice problems. For those of you who wish to dive even deeper into the subject, you can find a good collection of problems at the end of each chapter of the print edition of the book, whose solutions are available online.

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