Math 361 X1, Spring 2003

Extracredit Projects

Ground rules

• EC credit: Extra credit will be awarded in the form of up to 10 exam points, or three additional dropped HW scores. The amount of credit awarded depends on the size and difficulty of the project, and the amount of work done. The work required for full EC credit should be the equivalent of three regular HW assignments.
• Projects: Below are some possibilities for EC projects. Some of these are mainly programming projects, while others require reading some literature, or doing research on the web. Most projects are open-ended in the sense that there is no way to predict the final outcome and set a specific goal at the outset; the goals that can be accomplished will emerge as a project gets underway. For this reason it is essential that you meet with me at least once during the course of the project to discuss progress and directions to take.
• Group work: Although the projects are primarily intended as individual projects, some are suitable for group work (for instance, if they involve, a theoretical and programming component), so I may consider, on a case by case basis, work in groups of two, but check with me first. It goes without saying that group projects require all parties to contribute their share to the project.
• Final report: All projects should include a written report, along with the source code for any programs written for these projects (see below). The length of this report depends on the nature of the project, so check with me first. A neatly handwritten report is okay, though I would prefer receiving the report in electronic form. Acceptable formats for an electronic version are ascii, tex, latex, mathematica notebooks and their equivalents in maple and matlab (but not MS Word, Powerpoint, and the like); in either case, also give me a hard copy printout.
• Programs: Most projects involve some amount of programming. Pick a language you feel comfortable with and which is appropriate for the task; most likely, it will be one of C/C++, mathematica, maple, perl, java. The programs should be portable and compile in a Solaris environment. If you write in java, create a webpage containing the program. For other languages, send me the source code, so I can compile the program myself.
• Academic honesty: An EC project is not intended as a cheap way to make up for missed homework assignments. Since a project is worth up to the equivalent of three regular HW assignments, the amount of time invested in such a project should be comparable to that needed to complete three HW assignments. The work on the project should be your own work, not something that you may have copied from sources on the web, or elsewhere. Also, the project should not be something you have done for another class. (Of course, it's perfectly okay to re-use code fragments.) Some projects deal with well-known problems, so it is possible that there is material out there that I am not aware of. If you find something appropriate, let me know, so we can adjust the goals accordingly.
• Deadlines:
  • Monday, April 7: Last day to take on a project. This requires meeting with me in person.
  • Monday, April 21: Last day to meet with me to report on the progress of your work. If you have not started with the project by then, the project will not count.
  • Monday, May 5: Last day to turn in a project.

Projects

• The superbowl stock market predictor. This indicator predicts that the stock market will go up in a year when the NFC champion wins the Super Bowl, and will decline if the Super Bowl is won by the AFC champion. According to this article from Forbes Magazine, this indicator has been correct 29 out of the last 36 years. A naive analyis (modelling the stock market performance by a sequence of coin tosses) suggests that such a success rate is extremely rare. However, I suspect that the assumptions underlying this analysis are not appropriate. Is there a more realistic model under which the observed success rate (29 out of 36) is not that unlikely to have come up by pure chance? (I have some ideas in that direction, but they would have to be confirmed by data.) The superbowl predictor has been around for many years (I first heard of it in the mid 80s), so I suspect that there is quite a bit of material out there, in mainstream media, and perhaps also in the statistical literature. The main task of this project would be to try to dig out relevant sources through searches on the web, or/and on electronic databases accessible through the UIUC library, or printed materials, study the literature found, and summarize the results in a report.
• Lotteries in the real world. In its simplest form, a standard lottery is easy to analyze, using a box/ball model, and the probabilities for matching a given number of the lottery numbers drawn can be precisely determined. Also, if the the prizes to be won are fixed, then the expected win at a single drawing, and hence the fair price of a lottery ticket, can easily be computed. However, things get more complicated if the prizes are variable and, for example, involve a "jackpot" that accumulates if there are no winners at a given drawing. Occasionally, such jackpots reach almost astronomical numbers and then make headlines and cause a big rush at gas stations and grocery stories by people wanting to buy tickets. This is a situation that has been analyzed for the Powerball Lottery (which is not available in Illinois) in this article by Laurie Snell and Bill Peterson. The project would require studying this and other sources, perform a similar analysis for the lotteries available in Illinois, and summarize the findings in a report.
• The birthday problem with real data. Here is a fun project. The birthday problem deals with the probability that, in a group of a given size, at least two share a common birthday. A more general question is this: Given n and k between 1 and n, what is the probability that in a group of n people there are exactly k distinct birthdays in this group? This question can be answered theoretically (see the EC problem in HW 5), or numerically via computer simulation. In any case, it would be interesting to test these theoretical values against sets of real birthdays. The data sets would have to be publicly available, preferably in some form that is easy to use as data set of a program. (You don't want to manually key in hundreds of birthdays.) One example would be birthdays (or days of death) of U.S. presidents, another would be birthdays of famous mathematicians (here is a possible source for these dates), but there are many others.
• Generalizations of the birthday problem. There exist many programs, including java applets, simulating the standard birthday problem (which asks for the probability of a matching birthday in a group of a given size), so writing such a program would not be suitable for an EC project. However, there are a number of related questions which have received much less attention. Many of these are too difficult to solve theoretically, so it would be interesting to approach these questions via computer simulations. Here are some examples: (1) Given n and k between 1 and n, let f(n,k) be the probability that there are exactly k distinct birthdays in a group of n people. Determine f(n,k) experimentally, via simulation; then for given n, try to find the value of k that maximizes this probability, say kmax(n), and try to determine (experimentally) how kmax(n) behaves as a function of n. (2) The original birthday problem shows that in a group as small as about 23 people, with probability greater than 1/2 there is a duplicate birthday. How many people does it take so that there is a greater than 50/50 chance of a triplicate birthday? More generally, consider this question with "triplicate birthday" replaced by "k-fold" birthday, i.e., a day that is the birthday of k people.
• Random walk: Consider the two-dimensional integer lattice with points (n,m), where n and m are integers. A random walk is a path along the nodes of this lattice such that, from any given point, the path moves one unit up, down, left or right, with equal probability (i.e., 1/4). It is known that, with probability one, a random walk starting at the origin eventually returns to the origin, but one can ask a number of related questions, for example: (1) For each n, find the probability f(n) that the random walk returns to the origin in exactly n steps. (2) What is the number of steps n that maximizes this probability? (3) What is the expected number of steps needed to return to the origin? Furthermore, one can formulate anologous questions for random walks in 3 or more dimensions. Here things get more interesting, as in dimension 3 or greater there is a positive probability that a random walk never returns to its original position. These questions are too difficult to completely analyze theoretically, so the main goal of the project would be to write computer simulations to come up with experimental answers.

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Last modified Mon 03 Mar 2003 01:39:07 PM CST