## Homework Measure Theory And Statistics

## STA711: Probability & Measure Theory

### Syllabus

Week | Topic | Homework | |
---|---|---|---|

I. Foundations of Probability | Problems | Due | |

Aug 25-27 | Probability spaces: Sets, Events, and σ-Fields | hw1 | Sep 03 |

Sep 01-03 | Construction & extension of Measures | hw2 | Sep 10 |

Sep 08-10 | Random variables and their Distributions | hw3 | Sep 17 |

Sep 15-17 | Expectation & the Lebesgue Theorems | hw4 | Sep 24 |

Sep 22-24 | Inequalities, Independence, & Zero-one Laws | hw5 | Oct 01 |

Sep 29-01 | Review & in-class Midterm Exam I ('12, '13) | Hists: | Exam, Course |

II. Convergence of Random Variables and Distributions | |||

Oct 06-08 | Convergence: a.s., pr., L_{p}, L_{oo}. UI. | hw6 | Oct 22 |

--- Fall Break (Oct 11-14) --- | |||

Oct -15 | Expectation Inequalities | ||

Oct 20-22 | Strong & Weak Laws of Large Numbers | hw7 | Oct 29 |

Oct 27-29 | Fourier Theory and the Central Limit Theorem | hw8 | Nov 05 |

III. Conditional Probability & Conditional Expectations | |||

Nov 03-05 | Cond'l Expectations & the Radon-Nikodym thm | hw9 | Nov 10 |

Nov 10-12 | Review & in-class Midterm Exam II ('12, '13) | Hists: | Exam, Course |

Nov 17-19 | Introduction to Martingales (a, b) | hw10 | Nov 24 |

Nov 24 | Heavy tails and Extreme Values | ||

--- Thanksgiving Break (Nov 26-30) --- | |||

Dec 08 | Review for Final Exam, Mon 1:25-2:40 | ||

Dec 14 | 2-5pm Sun: In-class Final Exam ('12, '13) | Hists: exam, course |

### Description

This is a course about random variables, especially about their convergence and conditional expectations, motivating an introduction to the foundations of modern Bayesian statistical inference. It is a course by and for statisticians, and does not give thorough coverage to abstract measure and integration (for this you should consider Math 632, Real Analysis: F09 vsn). Students wishing to continue their study of probability following Sta 711 may wish to take any of MTH 641 (Advanced Probability), MTH 545 (Stochastic Calculus), or STA 961 (Stochastic Processes). Students are expected to be well-versed in real analysis at the level of W. Rudin's *Principles of Mathematical Analysis* or M. Reed's *Fundamental Ideas of Analysis*— the topology of R^{n}, convergence in metric spaces (especially uniform convergence of functions on R^{n}), infinite series, countable and uncountable sets, compactness and convexity, and so forth. Try to answer the questions on this analysis diagnostic quiz to see if you're prepared. Most students who majored in mathematics as undergraduates will be familiar with this material, but students with less background in math should consider taking Duke's Math 531, Basic Analysis I: F09 vsn (somewhat more advanced than Math 431, Advanced Calculus I, but that's a good second choice) before taking this course. It is also possible to learn the background material by working through one of the standard texts (like the two listed above) and doing most of the problems, preferably in collaboration with a couple of other students and with a faculty member (maybe me) to help out now and then. More advanced mathematical topics from real analysis, including parts of measure theory, Fourier and functional analysis, are introduced as needed to support a deep understanding of probability and its applications. Topics of later interest in statistics (*e.g.*, regular conditional density functions) are given special attention, while those of lesser statistical interest may be omitted.

Most students in the class will be familiar with undergraduate-level probability at the level of STA 230. While this isn't required, students should be or become familiar with the usual commonly occurring probability distributions (here is a list of many of them).

Some weeks will have lecture notes added (click on the "Week" column if it's blue or green). This is syllabus is *tentative*, last revised , and will almost surely be superseded— reload your browser for the current version.

### Note on Homework:

This is a demanding course. The homework exercises are difficult, and the problem sets are long. The only way to learn this material is to solve problems, and for most students this will take a substantial amount of time outside class— six to ten hours per week is common. Be prepared to commit the time it will take to succeed, and don't expect the material to come easily. Working with one or more classmates is fine; but write up your own solutions in your own way, don't copy someone else's solutions (that's plagiarism). Homework problems are awarded zero to three points each, based on your success in *communicating a correct solution*. For the full three points the solution must be clear, concise, and correct; even a correct solution will lose points or be returned ungraded if it is not clear and concise. Neatness counts. Consider using LaTeX and submitting your work in pdf form if necessary (it's good practice anyway). Homeworks may be submitted in paper form (just bring them to class on Wednesday) or electronically in the form of pdf files (submit as Sakai "Assignments" or send by e-mail to the course TA, *jonathan.christensen*@stat.duke.edu).

### Note on Exams:

In-class Midterm and Final examinations are closed-book and closed-notes with one 8½"×11" sheet of your own notes permitted. Tests from recent STA711 offerings are available to help you know what to expect and to help you prepare for this year's tests:

Solutions are not made available for these, because many students can't resist looking up the answer when they get stuck and then the exams lose their value. If you bring your solutions to the TA or to me during our office hours we'll be glad to give feedback, hints, etc.### Note on Sakai:

Use your Duke University uid and password to log into the Sakai learning management system to see scores for your homework assignments and examinations. You may also submit homework as "Assignments" in Sakai. At present those are the only features of Sakai we are using, but if interest warrants we may use some others such as "Forums". It's a good idea to check your homework and exam scores in Sakai every now and then, to make sure your scores were recorded correctly.### Note on Evaluation:

Course grade is based on homework (20%), in-class midterm exams (20% each), and final exam (40%). Most years grades range from B- to A, with a median grade near the B+/A- boundary. Grades of C+ or lower are possible (best strategy: skip several homeworks, skip several classes, tank an exam or two), as is A+ (given about once every two or three years for exceptional performance). Your current course average and class rank are available at any time on request.### Note on Enrollment:

Spaces in this course are reserved for PhD students from the Statistical Science and Mathematics Departments. While other well-prepared students are welcome, space in the course is limited and in some years it is over-subscribed. Early applicants and participants in Statistical Science MS programs have the best chance of enrolling. Occasionally one or two exceptionally well-prepared undergraduate students wishes to take the course; there is a surprisingly cumbersome process for obtaining permission for that described on the Trinity College website.### Note on Auditing:

Unregistered students are welcome to sit in on or (preferably) audit this course if:- There are enough seats in the room, and
- They are willing to commit to active participation:
- turn in about a third or so of the homeworks (or a few problems on each of most HW assignments)
- take either the final exam or a midterm
- come regularly to lectures, and ask or answer questions now and then.

*all*students to participate actively. It hurts the class atmosphere and lowers students' expectations when some attenders are just spectators. I try to discourage that by asking active participation of everyone, including auditors, to make the classes more fun and productive for us all. Past experiences suggests that most auditors stop attending midway through the semester, when they have to balance competing demands on their time; if this course material is important to you, it is better to take the class for credit.

### Note on Absence:

No excuse is needed for missing class. Class attendance is entirely optional. You remain responsible for turning in homework on time and for material presented in class that is not in the readings. Try not to get sick at scheduled examination times.### Academic Integrity

You may discuss and collaborate in solving homework problems, but you may not copy— each student should write up his or her solution. Cheating on exams, copying or plagiarizing homeworks or projects, lying about an illness or absence and other forms of academic dishonesty are a breach of trust with classmates and faculty, and will not be tolerated. They also violate Duke's Community Standard and will be referred to the Graduate School Judicial Board or the Dean of the Graduate School.

Instructor:

Office:

Office Hour:

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Course Assistant:

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This course is the first of a three-quarter sequence in measure-theoretic probability. It is meant for students with a solid grounding in real analysis. The main topics to be covered (not necessarily in the order listed) will be :

- Existence and Extensions of Measures (1 week)
- Independence; Borel's SLLN (1 week)
- Measurable Functions, Transformations, and Random Variables (1 week)
- Integration and Expectation (1-2 weeks)
- Sums of Independent Random Variables (2 weeks)
- Birkhoff's Ergodic Theorem(1 week)
- Weak Convergence (1 week)
- Wigner and Marchenko-Pastur Theorems (1 week)

There will be weekly homework, a midterm and a final exam. Detailed lecture notes will be posted as the course progresses.

- Probability and Measure by Patrick Billingsley
- Probability: Theory and Examples by Richard Durrett

The midterm exam will held on Friday, February 3 at the regular class time. You may bring 1 page of notes.

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