Math 60850 - Probability
Spring 2016
Instructor: David Galvin
NOTE: all course information announced here is subject to change before the first day of semester! The date for the in-class midterm exam is tentative, and also subject to change beyond the first day of semester (though plenty of notice will be given if it changes).
About the course
Probability theory in the discrete setting (finite or countable outcome spaces) does not require much technical machinery --- once a probability is assigned to each possible outcome, the probability of landing inside some arbitrary subset of outcomes can be unambiguously declared to be the sum of the probabilities of the outcomes in that subset, and everything goes through nicely.
The story is very different when the space of outcomes is uncountable. For example, suppose one wants to model selecting a point at random from the interval unit interval [0,1]. It's clear that one should assign probability b-a to the event that the selected point lies in the subinterval [a,b]. But what about the probability that it lies in some much more complicated subset, like the Cantor set? A great deal of care has to be taken, mainly using the machinery of measure theory, to get things to behave well in this setting.
This course serves as an introduction to rigorous, measure-theoretic based probability theory.
Topics to be covered include:
- construction of probability triples to model, for example, selecting a point uniformly from an interval, and tossing a coin infinitely often;
- random variables, independence and expectation (which, in the measure theory setting, does not need an artificial distinction between ``discrete'' and ``continuous'' random variables);
- probabilistic inequalities, modes of convergences, limit theorems such as the weak and strong laws of large numbers and the central limit theorem;
- conditional probability and expectation;
- and other topics as time and interest allow, such as Martingales and Markov chains.
Math 60350 (Real Analysis) is a prerequisite for this course. If you have any questions, please contact me! (at the email address below).
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Basic information
- Meeting times: Monday, Wednesday, Friday, 3pm-3.50pm, Hayes-Healy 231, January 13 to April 27.
- Instructor: David Galvin, 132 Hayes-Healy (dgalvin1 at nd.edu). Feel free to email me anytime. I try to respond quickly to any question or comments.
- Office hours: Fridays, 11.30am-12.30pm.
- Textbook:
I'll follow the book A first look at rigorous probability theory (2nd edition) by Jeffrey Rosenthal, ISBN 981-270-371-3, and I'll probably also draw on Mathematics of Probability (Graduate Studies in Mathematics)
by Daniel Stroock, ISBN 147-040-907-0. Other books that are worth referring to include:
- R. Durrett, Probability: Theory and Examples
- R. Billingsley, Probability and Measure
- S. Varadhan, Probability Theory (Courant Lecture Notes)
- W. Feller, An Introduction to Probability Theory and its Applications (2 volumes).
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Assessment
- Homework: Written homework will be assigned roughly every other week; in total, homework will count for 100 points out of 450 (each homework equally weighted). Specific homework policies will be announced with the first homework.
- Quizzes: In-class quizzes will be given roughly every second week; in total, quizzes will count for 100 points out of 450 (each quiz equally weighted). Detailed information about each quiz (the material being covered, and when it will be given) will be announced in class a few days before each one.
- Midterm exam: There will be one in-class midterm, tentatively planned for Monday, February 29; this will count for 100 points out of 450. Detailed information about the midterm (such as the material being covered) will be announced in class in late February.
- Final exam: There will be a (cumulative) final exam for this course, on Wednesday, May 4, from 1.45pm to 3.45pm. It will count for 150 points out of 450. Specific information about the final exam, such as where it will be held, and what to do in the case of a conflict, will be announced in class during the final week of the semester.
- Final grade: A total of 414 out of 450 will earn you an A/A-; a total of 360 out of 450 will earn you a B+/B/B-; a total of 292.5 out of 450 will earn you a C+/C/C-.
- Sakai: Your marks on each of the graded components will be periodically updated on Sakai.
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Late assignments
Detailed solutions to homeworks and quizzes will be posted here on their due dates, so late work cannot be accepted. All homework must be done by the due date to receive credit, and all quizzes and exams must be taken at the assigned times. I will not consider requests for homework extensions, or make-up quizzes and/or exams, except in the case of legitimate, university-sanctioned conflicts. It is your responsibility to let me know the full details of these conflicts before they cause you to miss an assignment! Excepting university-sanctioned conflicts, it is your responsibility to be in class for all scheduled lectures.
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Homework
The fortnightly homework is an integral part of the course; it gives you a chance to think more deeply about the material, and to go from seeing (in lectures) to doing. It's also your opportunity to show me that you are engaging with the course topics.
Homework is an essential part of your learning in this course, so please take it very
seriously. It is extremely important that you keep up with the homework, as if you do not, you may quickly
fall behind in class and find yourself at a disadvantage during exams and quizzes.
You are permitted, in fact encouraged, to work together and help one another with homework, although what you turn in should be written by you. Providing detailed arguments in your homework is important, since learning how to write mathematics in a rigorous and yet concise and readable way is an essential part of graduate school in mathematics.
- Homework 1: Rosenthal 2.7.1, 1.3.5, 2.7.7, 2.3.16, 2.7.4. Solutions.
- Homework 2: Rosenthal 2.7.21, this, Rosenthal 3.2.2, 3.6.3 b), 3.6.5, 3.6.7. Solutions.
- Homework 3: Rosenthal 4.3.2, 4.3.3, 4.3.4, show that the correlation Corr(X,Y) is always between 1 and -1, Rosenthal 4.5.3, 5.5.1, 5.5.9. Solutions.
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Quizzes
Quizzes will be posted here in a single file that will be updated throughout the semester. Quizzes will not be totally problem-oriented, but rather will test basic understanding of definitions and theorems. Quiz solutions will also be posted here.
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Exams
Here is the midterm exam and here are solutions.
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Supplementary material
Here is where I will post any supplementary material for the course.
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Conduct
Honor code: This hardly needs to be said, but there's no harm in it: I expect all students to abide by the university's Honor Code pledge, to not participate in or tolerate
academic dishonesty. For this course, that means that although you may (and should) discuss assignments
with your colleagues, you must write the final version of each of your assignments on your own; if you use
any external sources to assist you (such as other textbooks, computer programmes, etc.), you should cite
them clearly; your work on the mid-semester exam and the final exam should be your own; and you will adhere
to all announced exam policies.
Class conduct: The lecture room should be a place where you should feel free to engage in
lively discussion about the course topic; don't be shy! But non course related interruptions should
be kept to a minimum. In particular, you should turn off or switch to silent all phones, etc.,
before the start of class. If for some good reason you need to have your phone on during class, please
mention it to me in advance.
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