Math/Stat 431 Introduction to the Theory of Probability - Lecture 1

Spring 2017, Lecture 1

Meetings: MWF, 9:55-10:45 AM Van Vleck B123

Instructor: Daniele Cappelletti

Office: 415 Van Vleck.

Office Hours: Thursday 9.00 - 10.30am, and by appointment.

E-mail:cappelletti@math.wisc.edu

This is the course homepage that also serves as the syllabus for the course. Here you will find our weekly schedule, and updates on scheduling matters. The Mathematics Department also has a general information page on this course.

I will use the class email list to send out corrections, announcements, etc.  Please check your wisc.edu email regularly.

Course description

Math 431 provides an introduction to the theory of probability, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include the axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.

From a broad intellectual perspective, probability is one of the core areas of mathematics with its own distinct style of reasoning. Among the other core areas are analysis, algebra, geometry/topology, logic and numerical analysis.

To go beyond 431 in probability you should take next 521 -- Analysis, and after that one or both of these: 632 Introduction to Stochastic Processes and 635 Introduction to Brownian Motion and Stochastic Calculus.  Those who would like a proof based introduction to probability could consider taking  Math 531 - Probability Theory (531 requires a proof based course as a prerequisite). 

Where is probability used?

Probability theory is used ubiquitously throughout the sciences and industry.  For example, in biology many models of cellular phenomena are now modeled probabilistically as opposed to deterministically (this is the subject area that got me interested in probability). As for industry, many models used by insurance companies are probabilistic in nature (see: actuarial science). Also, many of the models used by the finance industry are also probabilistic in nature (google "Black-Scholes" to see an example). Thus, those wishing to go into finance need to have a solid understanding of probability.

Prerequisites

Textbook

• A first course in Probability, by Sheldon Ross
• Probability, by Jim Pitman.
• Probability, statistics, and stochastic processes, by Peter Olofsson.
• Probability and random processes, by Geoffrey Grimmett and David Stirzaker.

Evaluation

• Midterm exam 1: Wednesday, February 22nd, 7:15 PM - 8:45 PM, 1361 Chemistry. Let me know in due time if you need to take a make-up exam!
  
• Midterm exam 2: Thursday, April 13th, 7:15 PM - 8:45 PM, 1361 Chemistry. Let me know in due time if you need to take a make-up exam!
  
• Final exam: Thursday, May 11th, 7:45 AM - 9:45 AM, ROOM TBA.

• First quiz: January 23, Monday, end of class.
  
• Second quiz: January 30, Monday, end of class.
  
• Third quiz: February 6, Monday, end of class.
• Fourth quiz: February 13, Monday, end of class.
  

The purpose of the quizzes is to give you practice answering probabilistic questions in a timed environment. We will have these quizzes until I decide they are no longer needed.  No makeup quizzes will be given, instead you may drop your lowest quiz score. 

Homework

Homework assignments will be posted on the Learn@UW site of the course. Weekly homework assignments are due Fridays at the beginning of the class. Note that there is a (short) homework assignment due on the first Friday (September 9).

Instructions for homework

• Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions.
• Homework is collected at the beginning of the class period on the due date. No late papers will be accepted. You can bring the homework earlier to the instructor's office or mailbox.
  
• Working in groups on homework assignments is strongly encouraged; however, every student must write their own assignments.
  
• Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible. If the answer is a simple fraction or expression, a decimal answer from a calculator is not necessary. But for some exercises you need a calculator to get the final answer.
• Answers to some exercises are in the back of the book, so answers alone carry no credit. It's all in the reasoning you write down.
• Put problems in the correct order and staple your pages together.
• Do not use paper torn out of a binder.
• Be neat. There should not be text crossed out.
• Recopy your problems. Do not hand in your rough draft or first attempt.
• Papers that are messy, disorganized or unreadable cannot be graded.
• I strongly encourage you to type up your solutions (perhaps using Latex).

Succeeding in this course: This course will be more difficult than previous math courses, although I strongly believe it will be more rewarding for those that put in the necessary effort. To succeed you should read the sections of the text before each lecture and then again after. You should work together in small groups and discuss the material.  (You may use Piazza to put a group together.) Such discussions are an invaluable way to learn the material. After working in the groups, you should carefully and clearly write-up your solutions making sure you actually understand the material. If, even after completing a homework assignment, you feel you still do not fully understand the material as well as you could (perhaps because you could not follow the discussion in your working group well), you should do more problems.

Webwork

During the course you will be asked to do some exercises on an online platform called Webwork. You can find it here: http://webwork.math.wisc.edu/webwork2
You will be provided with credential to access the relevant section. It may be a good idea to change your password once you have logged in.
To enter the solution, you just need to type it in. Webwork interface accepts numbers, fractions, functions, etc. , with the syntax that you would expect. While you do not have the possibility to explain your argument on Webwork, you can check whether your solution is correct, so you have an immediate feedback on whether you are on the right path. Also, if your Webwork solution is wrong, you can try again as many times as you like, until the due date. This can of course be helpful to you, as you may be able to assess what you understood on your own, and in case correct yourself before writing down the solution of the homeworks on paper.

Weekly schedule

Here is a tentative weekly schedule, to be adjusted as we go. The numbers refer to sections in lecture notes that can be found at the Learn@UW website.