by Jim Pitman
, Departments of Statistics
, The University of California, Berkeley.
The most current version of this statement is available online with hyperlinks at
Following are some remarks about my teaching philosophy, with reference to some courses I have taught at Berkeley in recent years.Statistics 20, Introduction to Probability and Statistics
I have taught this course many times using the text "Statistics" by Freedman, Pisani and Purves. While this text is excellent in many ways, its level of mathematical treatment is a bit too low for this course. I typically supplement the text with lectures which provide the conventional notation for data lists and random variables, and give the students some exercises in the formal properties of summation and expectation.
Statistics 134, Concepts of Probability.
This is a a one semester course aimed primarily at students who have done a year of calculus. Only a minority of these students, however, have any real facility with calculus. To ensure that all students leave the course with a comprehension of probability concepts unobscured by technical difficulties, in the first half of the semester I emphasize fundamental ideas of probability in a discrete setting, without reliance on calculus. Then I develop these ideas further using calculus tools in the second half of the semester. My experience is that students learn more from intuitive explanations, diagrams and examples than they do from theorems and proofs. In lectures I deal quickly with the main points of theory, then spend most class time on problem solving and applications of the main ideas. Perhaps most valuable thing for students to learn (and the hardest thing to teach) in a course like this, is how to pick up a probability problem in a new setting and relate it to the standard body of theory. The more students see this happen in class, and the more they do it themselves in exercises, the better they grasp the basic ideas. I developed a textbook for this course based on the above philosophy, (Probability
, Springer 1993) after many years of testing preliminary versions at Berkeley and other institutions. The book has stood the test of time, without requiring a new edition in over 15 years. Still, there are a number of ways in which this material might be further developed for the benefit of both students and instructors. One would be to work a significant computational component into the course, using the R
package as a tool to make such things as graphing densities, simulation, and numerical approximation more readily accessible to the student. An obstacle to such innovation is the lack of familiarity of students with R in particular and with computation in general. This might be corrected by making the course Statistics 133, Concepts of Computing with Data, a prerequisite for 134. Another innovation would be to make the existing database of problems and their solutions, currently accessible only to teaching assistants and instructors, directly accessible to students via the web. While I have been working towards this goal as a background project for some years, problems with data management and rendering of mathematics online have impeded progress. It appears however that the innovation in rendering of mathematics on the web provided by MathJax
, combined with the use of modern database and search tools, should soon allow provision of dramatically improved systems of this kind. So I hope to see progress on this in the next year or so. A third innovation I would like to make is to provide better connections between concepts as they arise in this course and online resources related to these concepts, such as articles in Wikipedia, with some rating of materials by their relevance to the course. Students might then be encouraged to explore developments of these concepts in applications and contexts too varied to be incorporated as part of the core material of the course. My current effort in this direction this can be seen at http://stat.regier.ws/books/probability_1_ed/
, which offers an online table of contents of my textbook with each topic linked to its appearance on a variety of other sources.Statistics 150, Stochastic Processes
This sequel to Statistics 134 offers students their first glimpse at the theory of stochastic processes and the richness of probabilistic reasoning. This is a course which can engage the best students and draw them into pursuing graduate studies in probability and stochastic processes. It is rewarding to see how often the best students in Stat 150 are motivated to go the next level of Statistics 205 to learn the full measure-theoretic justification of the various limit operations which must be taken for granted in the undergraduate treament.
For students not headed to grad school, it is not so clear how much they benefit from a course like this. I think these students would be better served by a more computation-intensive course involving extensive use of simulation to get a more hands-on feeling for what various stochastic processes are like, and how they model various phenomena of interest. I wish the department had the human resources available to help develop such a course, but sadly this seems unlikely in the present economic climate. Statistics 205, Probability Theory
This is a graduate level introduction to probability theory. There are now a variety of textbooks available for this course. I follow my own development of the core material with reference to Durett's textbook
for details and problems. The heart of this course is the weekly problem sets. Students typically find these challenging, as they often demand synthesis of ideas from various sources. Again, the emphasis on problem solving helps to actively engage the students in the material. Over the years, students have written up lecture notes from this course, most of which can now be seen on the course homepage: http://bibserver.berkeley.edu/205/
I am working on developing these notes into an interactive web-based resource, with hyperlinked cross references, which should be of great value as a teaching aid. As another resource for this course, I have developed a searchable database of problems from past PhD qualifying examinations in probability. Innovations in database management and presentation of math on the web should greatly increase the value of these teaching resources in the next few years.Expository Writing
I continue to value highly the creation of documents which survey a topic in a way which makes it more accessible to students.
In addition to lecture notes posted on the web, mentioned above, two efforts in this direction in the last few years, directed
towards beginning researchers in probability, are as follows:
Jim Pitman and Marc Yor
Itô's excursion theory and its applications
Japanese Journal of Mathematics Vol. 2 No. 1, 83-96 (2007).
Special Feature: Award of the 1st Gauss Prize to K. Itô
A presentation of Itô’s excursion theory for general Markov processes
is given, with several applications to Brownian motion and related
Alexander Gnedin and Ben Hansen and Jim Pitman
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws
Probability Surveys Vol. 4, 146--171 (2007).
This paper collects facts about the number of occupied boxes in the
classical balls-in-boxes occupancy scheme with infinitely many positive
frequencies: equivalently, about the number of species represented in
samples from populations with infinitely many species. We present
moments of this random variable, discuss asymptotic relations among
them and with related random variables, and draw connections with
which appears in various manifestations.