To many mathematicians, the computer has been an invaluable aid to understanding various kinds of mathematical behavior. The role of the computer can be brought into direct analogy with the performance of scientific experiments done in the laboratory. Here is typically the situation: the investigator believes some type of mathematical property is holding, which can be stated in the form of a conjecture. He then proceeds to devise and perform experiments on the computer which can substantiate his belief. He uses the computer interactively, with a language suitable and adaptable to easily construct programs involving quantitative and logical expressions. He then views the results. In most situations, the experiments suggest the investigator's beliefs to be incorrect, although they typically indicate what may actually be true. He then reformulates the conjecture, and either repeats the experimental process, or attempts to prove mathematically his conjecture, resulting in a theorem. Interactive computing is important. The investigator needs to create his programs and view the results in a timely fashion, in order to facilitate his thinking process and enhance his intuition.
The main purpose of this course is to expose the student to the concept outlined above of using the computer to gain insight into mathematical behavior. The course centers on certain aspects of probability, where the student can computationally ``create'' random phenomena in order to view them and investigate properties. The emphasis is on topics not usually taught in basic probability courses, in order for the experimental process to be fully appreciated. In several instances the underlying mathematical ideas are discovered only through computation.
The student is required to complete assignments on the computer using the language APL2. This interactive language ranks among the best for performing mathematical experiments, in that programs can be quickly devised, implemented, and changed, the results being readily available in order to facilitate the experimental procedure. In a course like this it is important to concentrate on one language and to have one class account since, for example, some of the assignments require the student to load to their accounts programs created by the instructor (``locked'' so that no one can see how it works) which generate output (sometimes different output for each student) the student must analyze. Help sessions will be given during extra hours for students not familiar with APL2.
An example of a typical assignment, given in conjunction with the topic on limit theorems, involves the creation of an insurance model, where the student simulates the daily assets of an insurance company with clients' accidents and claim sizes occuring randomly. Usually, the student is asked to discuss the computer results, in relation to his own intuition and what has been discussed in class.
Although techniques of random simulation, taught at the beginning, are discussed in several courses on campus, this course differs dramatically from them in that the techniques developed are used for the purpose of mathematical experimentation. They are also presented and formulated in their complete mathematical context, and the student will be required to thoroughly understand the concepts involved. For this purpose, some of the experiments actually test and compare the various techniques.
The course begins with a mathematical development of the basic tools needed to generate random phenomena on the computer, namely, creating first pseudo-random numbers uniformly distributed on (0,1) (3 weeks), focusing primarily on a complete understanding of how the language used for the course performs this task, and then random variables with arbitrary distributions (2 weeks). The course then proceeds in using the tools to perform experiments for the purpose of gaining mathematical insight into certain areas of probability. These include the theories and applications of: Monte Carlo method (2 weeks), limit theorems (which includes among several topics an application of insurance models, the limiting distribution of of order statistics, and the behavior of the eigenvalues of large dimensional random matrices) (6 weeks), and the theory of stochastic processes (2 weeks). The emphasis will be on random behavior not usually taught in a standard probability course, stressing the importance of the experiments.
There are typically four projects assigned throughout the semester.