# First Look At Rigorous Probability Theory !!EXCLUSIVE!!

A mathematically rigorous course in probability theory which uses measure theory but begins with the basic definitions of independence and expected value in that context. Law of large numbers, Poisson and central limit theorems, and random walks.

## First Look at Rigorous Probability Theory

Other textbooks that you can look at for a different perspective: Measure Theory by Cohn or Real Analysis: Modern Techniques and Their Applications by Folland for measure theory background. Probability and Measure by Billingsley, A Course in Probability Theory by Chung, A First Look at Rigorous Probability Theory by Rosenthal for the main subject material of the course.

This textbook is an introduction to probability theory using measure theory. It is designed for graduate students in a variety of fields (mathematics, statistics, economics, management, finance, computer science, and engineering) who require a working knowledge of probability theory that is mathematically precise, but without excessive technicalities. The text provides complete proofs of all the essential introductory results. Nevertheless, the treatment is focused and accessible, with the measure theory and mathematical details presented in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects. The text strikes an appropriate balance, rigorously developing probability theory while avoiding unnecessary detail.

I am very pleased that, thanks to the hard work of MohsenSoltanifar and Longhai Li, this solutionsmanual for my book1 is nowavailable. I hope readers will find these solutions helpful as youstrugglewith learning the foundations of measure-theoreticprobability. Of course, you will learn best if youfirst attempt tosolve the exercises on your own, and only consult this manual whenyou are reallystuck (or to check your solution after you think youhave it right).

My book has been widely used for self-study, in addition to itsuse as a course textbook, allowing avariety of students andprofessionals to learn the foundations of measure-theoreticprobability theoryon their own time. Many self-study students havewritten to me requesting solutions to help assesstheir progress, soI am pleased that this manual will fill that need as well.

for all Borel S R, proving that P (Y BX) = B gX(y)(dy) withprobability 1.Second, by considering E(1BX) = P (Y BX), thesecond equation follows from the first equationfor the special caseY = 1B. Now, by usual linearity and monotone convergence argumentsthe secondequality follows for general Y .

Text The course is centered around Rosenthal's A first look at rigorous probability theory, 2nd edition (ISBN 978-981-270-371-2). The book is well-regarded and relatively inexpensive (around $30). However, if purchasing the text presents a financial hardship for you, PLEASE let me know so we can see how to resolve the issue.

In absence of randomized controlled experiments, identification is often aimed via instrumental variable (IV) strategies, typically two-stage least squares estimations. According to Bayes' rule, however, under a low ex ante probability that a hypothesis is true (e.g. that an excluded instrument is partially correlated with an endogenous regressor), the interpretation of the estimation results may be fundamentally flawed. This paper argues that rigorous theoretical reasoning is key to design credible identification strategies, aforemost finding candidates for valid instruments. We discuss prominent IV analyses from the macro-development literature to illustrate the potential benefit of structurally derived IV approaches.

A graph is a collection of points with edges drawn between them. Graph theory was first introduced by Leonhard Euler in his solution to the Königsberg bridge problem in 1736. Since then, graph theory has become an active area of study in mathematics due both to its wide array of real life applications in biology, chemistry, social sciences and computer networking, and to its interactions with other branches of mathematics.

Network structures and network dynamics are a fundamental modern tool for modeling a broad range of problems from fields like economics, biology, physics, and sociology. Mathematical and machine learning techniques can be used to reveal underlying network structures. The course will use graphs (sets of nodes connected by edges) as a common language to describe networks and their properties. On the theoretical side, the course will cover topics such as basic probability, degree distribution, spectral graph theory (adjacency matrix, graph Laplacian), diffusion geometries, and random graph models. Applications will range over topics such as epidemics, marketing, prediction of new links in a social network, and game theory. The course will also include hands-on experiments and simulations. Three class meetings per week.

The p-adic numbers were first introduced by Kurt Hensel near the end of the nineteenth century. Since their introduction they have become a central object in modern number theory, algebraic geometry, and algebraic topology. These numbers give a new set of fields (one for each prime p) that contain the rational numbers and behave in some ways like the real numbers, but they also possess some unique and unexpected properties. The main objective of this course will be to rigorously construct the p-adic numbers and explore some of the algebraic, analytic, and topological properties that make them interesting. Four class meetings per week.

(Offered as STAT 360 and MATH 360) This course explores the nature of probability and its use in modeling real world phenomena. There are two explicit complementary goals: to explore probability theory and its use in applied settings, and to learn parallel analytic and empirical problem-solving skills. The course begins with the development of an intuitive feel for probabilistic thinking, based on the simple yet subtle idea of counting. It then evolves toward the rigorous study of discrete and continuous probability spaces, independence, conditional probability, expectation, and variance. Distributions covered include the binomial, hypergeometric, Poisson, normal, Gamma, Beta, multinomial, and bivariate normal. Other topics include generating functions, order statistics, and limit theorems.

This course serves as a first introduction to Lie groups and Lie algebras. We will examine the structure of finite dimensional matrix Lie groups, the exponential and differentiation maps, as well as compact Lie groups. We will also study Lie algebras, ideals and homomorphisms, nilpotent and solvable Lie algebras, Cartan subalgebras, semisimplicity and root systems, and the classification of semisimple Lie algebras. This classification is not only a fundamental result in Lie Theory, but is also an archetype of classifications that appear in other areas of math. More amazingly, this classification is embodied in simple combinatorial pictures called Dynkin diagrams, which underlie surprisingly disparate fields, such as geometric group theory, quiver representation theory, and string theory. Four class meetings per week.

Competitions, which can include individual and team sports, eSports, tabletop gaming, preference formation, and elections, produce data dependent on interrelated competitors and the decision, league, or tournament format. In this course, students will learn to think about the ways a wide variety of statistical methodologies can be applied to the complex and unique data that emerge through competition, including paired comparisons, decision analysis, rank-based and kernel methods, and spatio-temporal methods. The course will focus on the statistical theory relevant to analyzing data from contests and place an emphasis on simulation and data visualization techniques. Students will develop data collection, manipulation, exploration, analysis, and interpretation skills individually and in groups. Applications may include rating players and teams, assessing shot quality, animating player tracking data, roster construction, comparing alternative voting systems, developing optimal strategies for games, and predicting outcomes. Prior experience with probability such as STAT 360 may be helpful, but is not required.

This module is an introduction mathematical proof and its applications to probability. The module introduces the topics of set theory, counting, probability and expectation and looks at the methods of proof needed to produce fundamental results in these areas. At its core is the aim to allow students to develop logical arguments applied to sets and simple experiments involving probabilistic outcomes.

Class: Tuesday and Thursday, 9:00-10:20am, CFA 211Website: dschlimm/80-110spring01Instructor: Dirk SchlimmOffice: Baker Hall A60BPhone: 268-5737Email: dschlimm@andrew.cmu.eduOffice hours: By appointment. The best way to contact me is via email.Course description: Although we spend the great bulk of our mathematical education learning how to calculate in a variety of ways, mathematiciansrarely calculate anything. Instead they devote their time to clearly statingdefinitions, finding simple axioms, making conjectures about claims thatmight follow from these axioms, and then proving these claims of finding counterexamples to them. Although thinkers since Aristotle have devoted enormous time andenergy to developing a theory of mathematical reasoning, it is only inthe last century or so that a unified theory has

__emerged.In__this course, we not only consider the modern theory of mathematicalreasoning, but we also consider several case studies in which a problemis simple to solve with mathematical reasoning but almost impossible tosolve without it. For example, we consider how to compare the sizes ofinfinite sets, and how to solve the Monty Hall-Let's make a deal Problem. This allows you to get a feeling for the power of abstract reasoning, in particular when it's consequences are not so obvious at first sight, but can be validated__byexperience.By__learning a few facts about the evolution of mathematics from prehistory to modern times, and also by setting the relevant mathematical concepts into their historical context, you should be able to develop a basic understanding of the history of mathematics and of fundamental problems in the philosophy of mathematics.Goals:Identify some interesting facts about the historical development ofmathematics and be able to discuss it their impact onmathematical reasoning.Be able to define and give examples of basic mathematical concepts,like syntax, semantics, definition, axiom, valid argument, proof, mathematical induction, conditional probability.

Be able to explain in what sense (formal) mathematical reasoning isobjective and rigorous.

Practice expressing and communicating ideas in a clear way.

Have fun with mathematics!

Quizzes: A short quizconsisting of 1-5 questions concerning recent material presented in classwill be handed out once a week (usually on Thursdays) at the beginning of class. It that has to be answered in class (5 minutes in total). This provides you some information about what you are expected to know, and allows you to test your knowledge of the subject. Furthermore, it provides feedback to the instructor about the difficulties of the course material.Homework: Usually short assignments are assigned during on Thursdays and are due the following Tuesday. After each of the main topics (see the course overview) I will want you to write a short summary, but then you will have more time to complete it. Corrected assignments should be kept in the portfolio (see below). In order to obtain a better grade assignments can be redone and handed in on the next day of classes after they were handed back. This allowsyou to go over your work again and the opportunity to learn fromprevious errors. The assignments will be posted on the class web-site. It is your responsibility to obtain the assignment if you miss class. Attempting to give an excuseanywhere in the vicinity of: ``I didn't know there was an assignment,''or ``I missed class and my friend gave me the wrong assignment'' will causeexcessive irritation on the part of the instructor. You are free to collaborateon homework, but not to copy answers from friends. Assignmentsare due at the beginning of class on the date mentioned in the assignment,and have to be turned in on paper (except where explicitly mentioned).You may type them up or turn them in in legible handwriting. If you usea word-processor, make sure to use the spell-checker.Portfolio: Students are expected to keep a portfolio about thecontents of the course. This gives you the opportunity to organize the material presented in class in a neat and clear way. It will help you to keep track of where we are in the course. It will also make it easier for you to review the material and thereby help you to find out what you have really understood and what is not yet clear to you. The portfolio must be kept in a three-ring binder made for standard sizenotebook paper (8 1/2 by 11 inches, preferably not less than 1 inchwide at the back). Your name should be clearly writtenon the outside of the binder.You may use any type of paper you like, lined or unlined, and of anycolor it seems good to you to use. However, please use paper that has punchedholes correctly placed for insertion in the binder, and please use full-sizepaper.The portfolio should include:Table of contents.The pages of your notes and homework should be numbered (maybe separatenumberings, like H-1, H-2, for the homeworks).

For each lecture:

notes, containing the main concepts and ideas.These will be useful when reviewing the material, and for the summaries and essay you are expected to write.

a few remarks about what you find interesting or puzzling.Brieflyanswer the following questions: (a) What did you find most interesting in today's lecture? (b) What did you find most confusing in today'slecture? (c) What would you like to know more about that was mentioned in today's lecture? (d) Say one thought you had during today's lecture (does not have to be related directly to the material presented in the lecture).Please, don't write just one-word answers, but take a few minutes after each lecture to review the lecture. By answering these questions you learn how to reflect aboutthe material presented in class.

Homeworks assigned in class.

Index of technical terms introduced in the course (for quick reference).The index should onlycontain a sorted list of terms, or names, and have a reference to thepage where the term is defined or explained. Sometimes you may havemore than one reference. In case you don't have a definition already,you may look for one.Here's a list of terms I expect to find in the index at the end ofthe semester (by then you'll be familiar with all of them!):-,=,1-1 function,7-adic system,Archimedes,Aristotle,atomic sentence,Axioms of probability,Babylonians,base case,base clause,Bayes' Theorem,binary numbers,Cantor's diagonal argument,cardinality,commensurable,completeness,conclusion,Conditional probability,constructive mathematics,countable,De Morgan rule,deductive,deductively valid,definition of even,denumerable,Egyptians,entailment,Euclid,Eudoxos,expressively complete,fallacious ,final clause,form of an argument,formal,formal language,formula,function,Georg Cantor,George Boole,Gerhard Gentzen,Gottlob Frege,Hypatia,indirect,induction hypothesis,induction step,inductive,inductive clause,inductive/recursive definition,inference rule,injective,interpretation,Intuitionistic logic,irrelevant,Knights and Knaves,Kurt Gödel,Law of total probability,mathematical induction,molecular sentences,N,names of syllogism,natural deduction,neolithicum,non-classical logics,notation,numbers,objectivity,Plato,premise,proof,proof by contradiction,proof of irrationality of the square root of 2,proof that there are infinitely many prime,proposition,propositional formula,Pythagoras,Q,quantifier,R,reductio ad absurdum,semantic method to check validity of arguments,semantics,sentential/propositional logic,sound,soundness,statement,syllogism,syntax,tautology,terms in arithmetic,Thales,Thales' theorem,truth table,Z.

Reading material

All materials for this course should be kept in the portfolio binderat all times. Use dividers to mark off each section. Please arrange themin the order mentioned above.Remember: your portfolio is the embodiment of your work for this course.A complete and well presented portfolio virtually guarantees you a goodgrade. The opposite is also true.The portfolio has to be handed in once during the semester and at the end of the course. It will be returned with comments and a grade. After the final exam the portfolio has tobe picked up from the instructor's office. Only the end of semestergrade of the portfolio will count towards the final grade. The grade for theportfolio will be basedon completeness of content and clarity of exposition. What I will look for in particular is the following: Are all pages legible? Is the table of contents complete? Is there at least a page for each lecture? Are the main topics of each lecture summarized briefly? Are there personal remarks about interesting or puzzling points? Are all homeworks included (all versions of the ones which were redone)? (The content of the homeworks does not contribute to the portfolio grade, but to the grades for the homeworks.) Is the index complete? Are all terms in the index referencedcorrectly? Essay:A 3-10 page essay on a topic related to the course must be handed inby April 26.The topic can be chosen by the student, but must be approved by theinstructor. Every student must have picked a topic by April 12, and anoutline must be presented to the instructor by April 19. The essay provides you the possibility to study in more detail a particular subject of the course that interests you most. It should also contain a small technical part, like a brief proof.Note that thedates mentioned above are the deadlines, no extensionswill be granted. However, you can hand in the essay earlier if youlike, even before the midterm exam. Starting early with the essaygives you the advantage of having more time to work on it, discussit with the instructor, and allows you to avoid being cluttered withwork at the end of the semester.Exams: Two exams will be held during the course: on Tuesday,March 6, and Thursday, May 3. They will coverthe material up to the date of the exam. The final exam will be comprehensive.Class participation: Class participation is expected. This includesshowing up regularly (missing 1 or 2 classes is reasonable, missing 7 or8 is not -- if you have to miss a class, please tell the instructor why),showing up prepared, making an effort to answer questions posed, contributeto class discussions, and present small problems in class. It is a well-known fact that active learning (e.g., participating in discussions) is much more effective than passive learning (e.g., reading), thus you get more out of the course if you are actively involved.The reading material only supplements the lectures, but cannot compensate for them. Most material covered in class will not be available otherwise. And although class participation will not be explicitly graded, the quizzes, which cover material presented in class and which are administered in class, do count towards your final grade.Grading: The grade in this course depends on your continuouseffort during the semester. The final grade will be based on six componentsaccording to the following