419: Models of randomness in a variety of fields: actuarial studies, economics, biology, engineering, and others as appropriate for student population. Discrete time Markov chains, Poisson processes and generalizations, time series, Brownian motion, and dynamic programming. An important part of the course is an opportunity for a student to become involved in an actual modeling problem.
519: Models of randomness in a variety of applications. Discrete and Continuous Time Markov chains, Renewal processes and generalizations, queueing theory, time series, Brownian motion, and dynamic programming. Completion of basic linear algebra and probability is assumed.
419: Math 122 and at least one of Math/Stat 223, 311, 319, 360, 370
519: Linear algebra at the level of Math 122 and probability at the level of Math 360 is assumed.
Some experience using Excel, R, or Python will also be VERY helpful (or Matlab, or to a lesser extent Maple or Mathematica), but it is not strictly a prerequisite.
419 Follow-up courses: Math 436 Numerical Analysis, various statistics classes
519 Follow-up courses: various statistics classes
Format: in-person, rather than hybrid or online. Meetings:
Tue, Thu 4:00pm-5:15pm in Pray-Harrold 304
"Final Exam" (actually, presentations) schedule: Tue Apr 25, 3:30-5:00 A HALF HOUR EARLY
Math 419W: CRN 26568, 3 credit hours.
Math 519: CRN 26574, 3 credit hours.
Class meetings will be mostly interactive lectures, with some time to discuss homework.
Professor Andrew Ross
Pray-Harrold 515m
andrew.ross@emich.edu
http://people.emich.edu/aross15/
(734) 487-1658, but I strongly prefer e-mail instead of phone contact.
Math department main office:
Pray-Harrold 515
(734) 487-1444
Mon/Wed:
From To Event
10:30 11:00 office hours
11:00 11:50 Math 120 PH 321
11:50 12:30 office hours and lunch
12:30 1:45 Stat 360 PH 321
1:45 3:00 office hours
Tue/Thu:
9:30 10:30 grant meeting (Thursdays only)
10:30 11:00 office hours
11:00 11:50 Math 120 PH 321
11:50 12:30 office hours, lunch
12:30 1:45 Math 499 PH 321
1:45 3:00 office hours
3:00 3:30 research meeting
4:00 5:15 Math 419W/519 PH 304
Fri
11:00-12:00 every other week: bio research meeting
11:00-12:00 once/month: department Colloquium (students invited!)
12:30- 2:30 once/month: department meeting
I am also happy to make appointments if you cannot come to the general office hours. Please send me e-mail to arrange an appointment. However, I am not available when I am teaching other classes (see above).
The Mathematics Student Services Center (or "Math Lab") is also here to help you, in Pray-Harrold 411. Their hours are posted here. Please give them a call at 734-487-0983 or just drop by. However, very few tutors have taken Math 419W/519, so while it's a good place to work (meet with classmates, etc.) the tutoring might not be as good as it usually is for 100, 200, and 300-level classes.
Many assignments in this course will be in the form of papers, which I want to be well written. I will be providing you with as much discipline-specific writing help as I can. You may also find it helpful to consult with the Academic Project Center for help in tuning up your writing.
A pack of 3-by-5-inch notecards. At the end of many class sessions, I will ask you to write out your thoughts on the class, to provide me feedback on how things are going. You might write a one-sentence summary of the class session, then something about what the high point was (most important, coolest, or most clear) and what the low point was (least important, boring, or most-unclear-but-important-so-please-explain-it-better-tomorrow!) Another way to think about it: "What was the most important thing you learned today, and what question still remains in your mind?" A pack of 100 notecards costs roughly $1.00
Heavy use of a computer is required in this class. We will mostly use Excel and R/Rstudio, but Python is also acceptable. Many days, it will be best to have a laptop in class, but if you can't bring one, you can probably buddy up with someone.
We will use files from the Electronic CoursepackOur recommended (not required) textbook is "Introduction to Probability Models", by Sheldon Ross (no relation to your instructor), published by Academic Press, 11th edition (though you could save money by buying an earlier edition). I will put at least one copy on reserve in the math tutoring center (probably an older edition)
We will use the Canvas system. You are expected to keep an eye on your scores using the system, and get extra help if your scores indicate the need.
Here is a list of books that I have found interesting and related to math modeling. Perhaps some of them will strike your fancy, too. I own the ones that are starred (*) and can lend them to you. Others you will have to find at the library or on the usual Internet booksellers. Links are given to Amazon, but I do not specifically endorse them or any particular bookseller. Of course, if you like a book you can see what similar books the online bookseller recommends.
Our primary goal is to teach you to be a good (or great!) stochastic math modeler. To be a good modeler, you need:
We have a few secondary goals, which may be more or less applicable to your personal situation:
We will start by reviewing basic probability ideas. We will also learn how to simulate a variety of random variables using Excel or Matlab (your choice)--doing little simulations will help understand a fair amount of the theory we will learn.
Time Series are used for a variety of things in economics and the various sciences. This will be the most statistics-oriented part of the class.
Dynamic Programming is a method of optimizing one's decisions as they unfold in time. It often includes some model of randomness, because we don't know what the future will hold. It is also used in some pattern recognition problems, such as speech recognition and genomic searches/ DNA alignment.
After that, we will talk about Discrete Time Markov Chains (DTMCs), which are used to model a wide variety of phenomena, from people moving between socio-economic classes to babies learning where one word ends and the next begins. Then, we will talk about Poisson Processes, which are useful for modeling the arrival of demands (like phone calls or customers) or other time-based phenomena (radiation particles, asteroids, etc.)
We will also study Renewal Theory, in which many of the results are completely intuitive, but there is one important result (called the Inspection Paradox) that takes some getting used to.
Queueing Theory is the study of how long people (or items) have to wait to be served.
Reliability Theory is in the book, but we will not cover it in this course unless there is a demand for it and some extra time.
Brownian Motion is the basis of a lot of stock market models. It is essentially a random walk. We will also look at some generalizations.
Class# Date 2019 day unit Topic HW Assigned HW Due Project Item Due 1 01-08 Tue Newsvendor Overview; Newsvendor intro Get-to-know-you 2 01-10 Thu Newsvendor PMF, CDF, and EV; finish Newsvendor Reading Journal Papers (Newsvendor) Get-to-know-you 3 01-15 Tue DynProg Dynamic Programming DynProg Reading Journal Papers 4 01-17 Thu TimeSeries Time Series: Trends, residuals TS1:Trends DynProg 5 01-22 Tue TimeSeries Seasonality TS2:Seasonality TS1:Trends 6 01-24 Thu TimeSeries MA, AR, ACF, PACF TS3:Time Series Tutorial 7 01-29 Tue TimeSeries Cross-Corr; Time series wrap-up DTMC pre-reading TS2:Seasonality 8 01-31 Thu DTMC Markov Chains intro preview of Ch 4 HW DTMC pre-reading 9 02-05 Tue DTMC Vector-Matrix Multiplication; matrix powers; vector-matrix in Excel TS3:Time Series Tutorial 10 02-07 Thu DTMC Balance Equations; inventory example; Evaluating Info on the Web; Plagiarism 11 02-12 Tue DTMC steady-state; transient; Web of Science intro; Annotated Bibliography 12 02-14 Thu DTMC Irreducible and Not; Symbolic Steady State; Pseudo-Random; Common and Antithetic PRNG 13 02-19 Tue DTMC MDP; Hidden Markov; Levels of Concern in Revising Ch 4 HW Proposal 14 02-21 Thu DTMC Wrap up DTMC; renewal process testing Preread Ch 5 Annotated Bibliog. NA 02-26 Tue break week NA 02-28 Thu break week 15 03-05 Tue Poisson Ch 5 intro; selfish queueing; exponentiality testing; cdf and pdf live update preview of Ch 5 HW Ch 4 HW; Ch 5 Prereading 16 03-07 Thu Poisson Define and Simulate Poisson Process Full Draft 17 03-12 Tue Poisson Poisson splitting and combining; M/G/infinity; NHPP Ch 5 HW Peer Review 18 03-14 Thu Project Presentations Final Report 19 03-19 Tue Project Presentations 20 03-21 Thu Poisson; CTMC 2-dimensional Poisson processes; bank example; homogeneous; Ch6 CTMC grad students Ch 6 HW 21 03-26 Tue Ch7 Ch 7 Renewal Processes; inspection paradox Ch 5 HW 22 03-28 Thu Ch7 CLT for RP; Renewal Reward; alternating RP; insurance ruin Ch 7 HW grad Ch 6 HW 23 04-02 Tue Ch8 Ch 8 Queueing Ch 8 HW 24 04-04 Thu Ch10 Ch 10 Brownian Motion Poisson Assumptions Ch 7 HW Proposal 25 04-09 Tue Ch10 Geometric and Integrated Brownian Motion; multivariate Normal Ch 10 HW Ch 8 HW Annotated Bibliog. 26 04-11 Thu misc TBD 27 04-16 Tue misc TBD Poisson Assumptions Full Draft 28 04-18 Thu misc TBD, maybe some presentations Ch 10 HW Peer Review 04-23 Tue no class--other classes having finals 04-25 Thu Final presentations, HALF-HOUR EARLY Final Report
By the end of the course, students will be able to:
Regular attendance is strongly recommended. There will be material presented in class that is not in the textbook, yet will be very useful. Similarly, there are things in the textbook that are might not be covered in class, but are still very useful. If you must miss a class, arrange to get a copy of the notes from someone, and arrange for someone to ask your questions for you. If you are stuck on occasion without your usual child care, you may bring your child to class, and need not even get advanced permission (this is my personal policy--I don't know if EMU has a policy). Please be considerate to your classmates if your child becomes disruptive.
My lectures and discussions mostly use the document camera, along with demonstrations in Excel and other mathematical software. I sometimes have PowerPoint-like presentations, and I distribute electronic copies.
Homework will be assigned about once or twice a week. It will sometimes be a small problem set designed to help you understand the behavior of math models. Other times, it will involve writing up a little paper on an assigned topic. All homework should be typed.
Homework papers should be submitted on-line via Canvas, where they may be checked by TurnItIn.com or a similar service. This is partly to help keep you honest, and partly to help you learn acceptable ways to cite the work of others. A side benefit is that sometimes TurnItIn finds papers relevant to your work that you would not have found otherwise!
There will be no exams unless the class has trouble being otherwise motivated. If you would like an interesting project, you could create a final exam for this course, along with a writeup justifying why each question is appropriate, and of course a solution key along with rubric for grading incorrect answers.
Instead of a mid-term and a final exam, you will do a mid-term and a final project. Your results will be reported in a paper and a presentation to the class. The grade for each project is split into:
Final presentations will be made during the time slot reserved for the final exam. If there will not be enough time to do all final presentations, then posters, random selection, or point-auction may be used.
No scores will be dropped, unless a valid medical excuse with evidence is given. In the unfortunate event of a medical need, the appropriate grade or grades may be dropped entirely, rather than giving a make-up, at the instructor's discretion. You are highly encouraged to still complete the relevant assignments and consult with me during office hours to ensure you know the material.
Your final score will be computed as follows:
Once final scores are computed, we will use the following grading scale:
92.0 and above : A 88.0 to 92.0: A- 84.0 to 88.0: B+ 80.0 to 84.0: B 76.0 to 80.0: B- 72.0 to 76.0: C+, etc.
Whether you go into industry or academics, you will need to be able to write reports on the mathematical work you have done. Math 419 is designed to enable students to apply math modeling techniques to formulate and solve problems in applied mathematics/operations research. In this class, students learn how to present their findings in the format of a peer-reviewed scientific journal or technical report, and how to present their findings in the format of PowerPoint-type presentations. Of the final grade in Math 419, over 50 percent is based on the writing assignments. Students are provided with the tools to enable them to communicate successfully their modeling findings. They receive written and oral feedback on smaller, staged writing assignments, as well as opportunities for revision, providing them with the skills to improve their writing and excel at writing complete papers. Students will individually write two full-length math modeling papers and presentations (those with an interest in secondary education may substitute one lesson plan for a modeling paper). Students who successfully complete Math 419 have the ability to read critically and evaluate peer-reviewed journal articles and present their own research in the same format. As such, Math 419 meets the requirements of a Writing Intensive Course in the Major of the General Education program.
Side note:
Math 419W is distinct from Math 311W, Mathematical Problem Solving, because 419W projects focus more on applied work where a substantial part of the difficulty is figuring out what problem we want to solve-do we optimize today's operations, or our tactics for the next few months, or our long-term corporate strategy? Also, Math 419W projects often start with real-world data that students obtain from their workplaces. Formal mathematical proofs are only rarely a part of Math 419W, whereas they are a mainstay of 311W. Computer simulations, computations, and sensitivity analysis are important parts of most 419W projects, while they are not usually important in 311W. Math 419W tends to consider stochastic (random) phenomena, while 311W considers deterministic formulas.
In the last few semesters, I've asked my math modeling students to give advice to you, future math modeling students, based on their experiences in my course. Here are some of the highlights:
See any common themes?
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