Probability and Statistics
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These areas consider the use of numerical information to quantify
observations about events. The tools and development are clearly
mathematical; these areas overlap with analysis in
particular. On the other hand, the use of the ideas developed here is
primarily not mathematical areas so much as in the
applications to the sciences.
60: Probability theory is simply enumerative
combinatorial analysis when applied to finite sets; thus the techniques
and results resemble those of discrete mathematics. The theory comes into
its own when considering infinite sets of possible outcomes. This requires
much measure theory (and a careful interpretation of results!) More
analysis enters with the study of distribution functions, and limit
theorems implying central tendencies. Applications to repeated transitions
or transitions over time lead to Markov processes and stochastic processes.
Probability concepts are applied across mathematics when considering
random structures, and in particular lead to good algorithms in some settings
even in pure mathematics.
62: Statistics is the science of
obtaining, synthesizing, predicting, and drawing inferences from data.
Elementary calculations of mean and standard variation suffice to summarize
a large, finite, normally-distributed dataset; the field of Statistics
exists since data are not usually so nicely given. If we do not know all
the elements of the dataset, we must discuss sampling and experimental design;
if the data are not normal we must use other parameters to summarize them,
or resort to nonparametric methods; if multiple data are involved, we
study the measures of interaction among the variables. Other topics include
the study of time-dependent data, and the foundations necessary to
avoid ambiguity or paradox. Computational methods (e.g. for curve-fitting)
are of particular importance in applications to the sciences and engineering
as well as financial and actuarial work.
Fields which contribute significantly to the development of these include
28: Measure and Integration;
05: Combinatorics; and
65: Numerical Analysis.
Fields which make significant use of these include
85: Statistical mechanics;
90: Economics and operations research; and
92: Biology and behavioural sciences;
You might want to continue the tour with a trip through computer and information sciences.
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Last modified 1999/05/12 by Dave Rusin. Mail: email@example.com