# Math 493 - Fall 2019

Instructor: Dr. Silas Johnson
Class time: MWF 2:00-2:50, Crow 204
Office: Cupples I, room 107A
Email: silas@wustl.edu
Office hours: Monday 3:00-4:00 (your class only) and Tuesday 12:30-2:00 (shared with Math 131)

## Course Outline

This is an introductory course in mathematical probability. Major topics include:

• The basic laws of combinatorics and probability
• Conditional probability and independence (in many contexts)
• Discrete random variables, examples of discrete distributions, how they work, and how to use them
• Continuous random variables, examples of continuous distributions, how they work, and how to use them
• Joint (multivariable) probability distributions
• Expected value, variance, and related ideas
• The Central Limit Theorem and other related results

This is a proof-based course, in that we will prove things in class, and you will at times (though definitely not always) be asked to do so on homework. However, the course will not require the degree of rigor expected in, for example, a real analysis course for math majors.

Learning objectives for the course include:

• Use the principles of combinatorics to compute probabilities of events
• Work with discrete and continuous random variables in general (including joint distributions), and understand the differences between these two types
• Understand common distributions (both discrete and continuous), how they arise, and how to use them to model real-world problems
• Analyze and solve problems involving expected values
• Write and read proofs involving random variables and distributions, including the use of basic limit theorems
• Communicate your work to others, both in proof and non-proof contexts
• Compare and contrast different problems or topics, especially the properties and uses of different distributions

## Teaching and Learning Philosophy

While it is easy in a lecture course to sit back and watch math happen on the blackboard, remember that the only way to learn math is to do math. It is crucial to your success that you engage with the material both in and outside of class. We will spend as much class time as we can allowing you to learn actively, rather than passively, in a variety of ways. Be open to the fact that class may not always look like what you think a lecture has to look like!

It is also easy to get lost in a crowd in a large-ish class, but you will have better results if you approach learning as a community activity. (After all, if that weren't important, you could take this class online!) The best resource you have is not the textbook, old exams, homework problems, or office hours, but your classmates. Use them! Work together in and outside of class, form study groups, ask for and give help. Conversely, remember that you are the best resource they have too, and communicating your ideas to others is a great learning technique and a crucially important skill.

Finally, don't be afraid to make mistakes. Despite all our emphasis on grades, failure is a crucial part of the learning process, and you should not expect to get everything right the first time. It is also your responsibility to help create an environment in which your classmates can safely engage in productive failure.

General advice for upper-level math courses:

• Collaboration is just as important as it was in lower-level courses (see above).
• Practice your mathematical writing early and often. It is a skill that takes significant practice and is easy to overlook.
• Start homework sets early, and give yourself time to think deeply about them.
• Read a section or two ahead before class, and attempt a few problems. This shouldn't make class boring; rather, it makes class an opportunity to clarify your thoughts and get a different perspective.

## Textbook and Materials

Textbook: A First Course in Probability, 9th edition, by Sheldon Ross. Some homework problems will be assigned out of the textbook, so if you have an older or newer edition, check with a classmate to ensure that you are doing the correct problems. (No credit will be given for doing the wrong problems.)

Recommended Textbook: Introduction to Mathematical Statistics, 7th edition, by Hogg, McKean, and Craig. A little bit of content near the end of the course will come from this book, but you are not required to buy it; I will ensure that you have access to everything you need either online or via lecture notes. This is also the textbook for Math 494, so if you are certain you will take that class, you can buy the textbook now.

Calculators: You may use any calculator you want on homework, but no calculators are allowed on exams. Exams will be written so as to not require calculators.

Canvas is our course management system; grades and homework will be posted there, as well as occasional announcements. To access Canvas, go to mycanvas.wustl.edu.

Homework: There will be around 9-12 problem sets, due approximately weekly. Homework will be submitted through Crowdmark; I will send instructions later on how to do this. Your lowest homework score will be dropped when calculating your grade.

I strongly encourage you to collaborate on your homework assignments and discuss the problems with others. However, you must write your solutions yourself, and you may not copy from any other student. See the section "Teaching and Learning Philosophy" above for more thoughts on the role of collaboration in the learning process.

Exams: There will be three in-class exams, plus a final exam. Pending availability of a larger classroom, the dates of these exams are:

• Wednesday, September 25th
• Monday, October 28th
• Monday, November 25th
• Monday, December 16th, 3:30-5:30pm (final exam)
Since these exams take place during normal class time, you are expected to take them at the assigned time. If an emergency arises that will preclude your attendance, contact Dr. Johnson as soon as possible. Do not book winter break travel that will conflict with the final exam; do not book Thanksgiving break travel that will conflict with Exam 3 unless you plan to drop that exam (see below).

Exams may or may not allow formula sheets; details will be discussed at least one week before the first exam. Either way, exams will not require excessive memorization.

Accommodations: Students requiring accommodations for a disability during exams or otherwise should register with Disability Resources as soon as possible. Send your VISA (which you will receive from Disability Resources) to Dr. Johnson at least two weeks in advance of the first exam so your accommodations can be arranged.

Grading Scheme: Your grade will consist of 25% homework, 15% for each of 3 in-class exams, and 30% for the final exam. Your lowest homework grade will be dropped, and your lowest in-class exam score can be replaced by your final exam score (if your final exam score is higher).

• A+: Only given at instructor's discretion
• A: 90 and up
• A-: 85-90
• B+: 80-85
• B: 75-80
• B-: 70-75
• C+: 65-70
• C: 60-65
• C-: 55-60
• D: 50-55 (D+ and D- ranges determined later)
• F: below 50
Note that scores will not be rounded; a total grade of 84.99 is still a B+, but 85.00 is an A-.
These ranges may be adjusted if absolutely necessary, but only in your favor. That is, the minimum score for a particular grade can move down, but not up.

If you take the course on a credit/no credit (pass/fail) basis, you must earn at least a C- to pass.

## Useful Resources

Campus Resources

External Math Resources

• Wolfram Alpha is a great way to check your work. Do not use it, however, to do homework problems for you.
• Sage is a Python-based system intended as an open-source alternative to Wolfram Alpha, Mathematica, and similar systems.
• GNU Octave is an open-source alternative to Matlab.
• Khan Academy has been immensely popular with many of my students as a supplemental resource, but I don't know what resources they have appropriate to this course.

## Approximate Schedule

This schedule is only an estimate! If you miss class, please confirm with another student so you can make sure you catch up on the correct material.

Week Dates Sections covered Notes
1 8/26-8/30 1.1-1.5, 2.1-2.5
2 9/3-9/6 Review, 3.1-3.3 No class Monday
3 9/9-9/13 3.4, 4.1-4.2
4 9/16-9/20 4.3-4.7
5 9/23-9/27 4.8 Exam 1 in class Wednesday
Tentatively covers sections 1.1-1.5, 2.1-2.5, 3.1-3.4, 4.1-4.5
6 9/30-10/4 4.9, 5.1-5.2
7 10/7-10/11 5.3-5.7 Last day to change to pass/fail, Friday
8 10/16-10/18 6.1-6.2 No class Monday-Tuesday
9 10/21-10/25 6.3-6.5, 6.7
10 10/28-11/1 ch. 6 wrap-up, 7.1-7.2 Exam 2 in class Monday
Tentatively covers previous material plus 4.6-4.9, 5.1-5.7, 6.1-6.3
11 11/4-11/8 7.4-7.5
12 11/11-11/15 7.6-7.8 Last day to withdraw, Friday
13 11/18-11/22 8.1-8.3
14 11/25-11/26 none, exam only Exam 3 in class Monday
Tentatively covers previous material plus 6.4-6.5, 6.7, 7.1-7.2, 7.4-7.8, 8.1-8.2
15 12/2-12/6 8.4, other topics TBD
Exams 12/16 Final exam 3:30-5:30