# Probability Theory: A Concise Course

This book, a concise introduction to modern probability theory and certain of its ramifications, deals with a subject indispensable to natural scientists and mathematicians alike. Here the readers, with some knowledge of mathematics, will find an excellent treatment of the elements of probability together with numerous applications. Professor Y. A. Rozanov, an internationally known mathematician whose work in probability theory and stochastic processes has received wide acclaim, combines succinctness of style with a judicious selection of topics. His book is highly readable, fast-moving, and self-contained.The author begins with basic concepts and moves on to combination of events, dependent events and random variables. He then covers Bernoulli trials and the De Moivre-Laplace theorem, which involve three important probability distributions (binomial, Poisson, and normal or Gaussian). The last three chapters are devoted to limit theorems, a detailed treatment of Markov chains, continuous Markov processes. Also included are appendixes on information theory, game theory, branching processes, and problems of optimal control. Each of the eight chapters and four appendixes has been equipped with numerous relevant problems (150 of them), many with hints and answers. This volume is another in the popular series of fine translations from the Russian by Richard A. Silverman. Dr. Silverman, a former member of the Courant Institute of Mathematical Sciences of New York University and the Lincoln Laboratory of the Massachusetts Institute of Technology, is himself the author of numerous papers on applied probability theory. He has heavily revised the English edition and added new material. The clear exposition, the ample illustrations and problems, the cross-references, index, and bibliography make this book useful for self-study or the classroom.

## Probability Theory: A Concise Course

This book is a concise introduction to modern probabilitytheoryand certain of its ramifications. By deliberatesuccinctness ofstyle and judicious selection of topics, itmanages to be bothfast-moving and self-contained.

7. An electronic computer contains 1000 transistors. Supposeeach transistorhas probability 0.001 of failing in the course of ayear of operation. What is theprobability of at least 3 transistorsfailing in a year?

I happened to take an introductory course on probability and statistics on two different universities. In one they used a horrible book, and in the other they used a truly amazing one. It's rare that a book really stands out as fantastic, but it did.

This course is an introduction to the theory and practice of probability and statistics. The emphasis is on the language of probability statistics, including its essential ideas and concepts. We will discuss the foundations of probability theory, basic descriptive statistics, graphical representation of data, point and interval estimation, hypothesis testing, correlation, and regression analyses. Working with the R language will give you an opportunity to see how the concepts discussed in class are applied to the real data sets.

So far we've been getting a lot of feedback that there are many in the course who feel like they are struggling with the more mathematical problems on the homework. While it might seem really hard now, please try your best to get through it, because unfortunately this level of mathematical rigour is central to both applied and theoretical machine learning research. We recognize that it is very tough if you haven't taken any course that covered probability or statistics, and it has been a long time for many of you since you had to apply calculus and remember things like how to manipulate exponents.

The most important background knowledge you will need is an intuitive understanding of probability at a basic level. In my opinion, the best resources are probability review sessions from other courses that you can find on the web. Here's some that I think cover the basics very quickly and clearly:

At the end of the course the student will have better knowledge and understanding as she/he will: master the basics in probability theory; master the most basic foundations of statistical techniques like mean values and errors estimates, validation of hypotheses.The student will be able to apply knowledge and understanding and in particular she/he will be able to: compute mean values and errors of a given data set; validate a simple hypothesis (which boils down to YES/NO alternatives) within a given confidence level; pin down the basic steps in setting up a simulation (singling out the relevant degrees of freedom, choosing a representation of the latter as data, choosing and implementing an algorithm for the simulation dynamic).The student will be able to make judgements and in particular she/he will be able to: distinguish cases in which a problem can be directly simulated and cases in which a modeling phase is compelling, capturing the relevant degrees of freedom; understand whether the relevant degrees of freedom are to be looked for in the form of macrostates.The student will also have acquired communication skills as she/he will be able to: present her/his results in a clean, precise and concise way; present her/his results both synthetically as for the overall picture and analytically as for the most delicate points; argue her/his thesis in public, in particular acting in a team.Finally, the student will have acquired learning skills as she/he will be able to: understand whether numerical simulation solutions are due in the context of problems she/he will be facing in the context of future studies or work; progress in the study of solutions (e.g. algorithmic solutions) beyond what she/he has learnt in this course.

Short review of probability theory and statistics, with an emphasis on numerical techniques (probability functions generation, data analysis).The problem of validating a hypothesis will be treated with some attention to mathematical rigour.A large fraction of the course will be devoted to applications of Markov processes theory. Modeling of queues will be the main application of the formalism.Basics of percolation theory will be introduced as an example of how a simple model can model a variety of phenomena, in particular epidemiological models.

Roughly in the middle of term a self-evaluation test on probability theory and statistics will be recommended, subject to compatibility with teaching stop dedicated to these activities (alternatively a homework will be evaluated). That is mostly intended as self-evaluation, but if it is well done (and only in that case) it will be taken into account in the final grade (2 points added as a bonus).Oral exam to which the candidate is admitted after having delivered a report on a project. In due advance of the exam session, a project will be assigned to the student. It will be a natural completion of subjects worked out during the lessons, with a clear assignement of what the student is supposed to do: numerical simulations, working out a few analytical results completing results obtained during the course, comparison of expected results and results stemming from simulations, computations of errors. Students will hand over their written report at the latest 24 hours before the oral exam takes place. Discussing the report will be the starting point of the oral exam. Besides a clean presentation of the technical solutions adopted and of the motivations for them, during the exam it could be required to reproduce a few results (for example: showing a program at work, verifying the correctness of a program). Both the report (and the results obtained) and the oral discussion will contribute to the final grade.Exams will take place at the university premises. Once again, in case of worsening of the Covid-19 pandemic, we will move to Microsoft Teams meetings. Students will make sure they can share their screen. 041b061a72