Statistical Science
In this note we briefly survey the evolution of statistical science during the last century, paying particular attention to its principal motivations: its applications, the mathematical theories that underpin it, and more recently, the developments of ele
- PDF / 1,178,364 Bytes
- 9 Pages / 547.087 x 685.984 pts Page_size
- 94 Downloads / 146 Views
w. STROOCK *
1. Origins and Overview The subject which is presently called probability theory seems to have its origins in the 17th Century. Indeed, it was only in the 17th Century that the concept of luck or chance evolved from its classical interpretation in terms of divine intervention on behalf of a chosen (i.e., lucky) individual into its modem interpretation in terms of perceived randomness resulting from inherent uncertainty or imperfect information 1. However, once this evolution took place, the calculation of probabilities quickly attracted the attention of several brilliant minds. To mention a few: Bernoulli, Huygens, and, somewhat later, de Moivre and Laplace. Although many ingenious additional calculations were made during the 18th and 19th Centuries, it was only after Lebesgue introduced his integration theory that probability theory as we know it today became possible. Indeed, what has become the standard model for probability theory requires the existence and understanding of countably additive measures. Thus, first S. Ulam, in 1932, and shortly afterwards A. N. Kolmogorov, in 1933, based their axioms, what are commonly called the Kolmogorov axioms, of probability theory on Lebesgue's theory of measures and integration. Although it will mean that we will have to ignore many very intriguing and potentially exciting aspects of the subject (e.g., finitely additive probability theory and what is sometimes called geometric probability theory), I will restrict our attention in this article to topics which can be rigorously treated in terms of Kolmogorov's axioms. In fact, I will present only a very sparse sampling of the topics considered by modem probabilists working in the framework of those axioms. The aspect of probability theory which I am attempting to convey here is the role which it plays as a middleman. Namely, because probability theory is couched in a (often deceptively) suggestive language, it has become a ubiquitous tool for the construction of models which expose mathematically rigorous connections between otherwise disparate fields. That is, probability theory often provides a bridge between topics whose relationship is not apparent until they are viewedprobabilisticaUy. Sometimes these bridges span the gulf between apparently unrelated areas within mathematics itself. At other times these bridges are between world of mathematics and its complement. In either case, the essential virtue of probability theory is the intuitive appeal of its language. Unfortunately, just because that language provides an intuitive relation between problems, it
* The author was partially funded by support from NSF grant DMS9625782. 1 I leamed this bit of history during a conversation with Shlomo Sternberg who, at the
time, was teaching probability theory to undergraduates.
B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
1106
D. W.
STROOCK
is only occasionally sufficient to provide a mathematically rigorous solution to the problems it relates. Be
Data Loading...
