| Meetings: MWF 11:00am-11:50am in 5231 Sewell Social Sciences |
| Instructor: Benedek Valkó |
| Office: 409 Van Vleck. |
| Office Hours: Tu 3:45pm-4:45pm, Fr 2:30pm-3:30pm or by appointment |
| E-mail: valko at math dot wisc dot edu |
This is the course homepage. Part of this information is repeated in the course syllabus that you find on Canvas. Important deadlines from the Registrar's page.
Math 431 is an introduction to the theory of probability,
the part of mathematics that studies random phenomena. We model
simple random experiments mathematically and learn techniques for
studying these models. Topics covered include axioms of
probability, random variables, the most important discrete and
continuous probability distributions, expectations, moment
generating functions, conditional probability and conditional
expectations, multivariate distributions, Markov's and Chebyshev's
inequalities, laws of large numbers, and the central limit
theorem.
Math 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.
From a broad intellectual perspective, probability is one of the core areas of mathematics with its own distinct style of reasoning. Among the other core areas are analysis, algebra, geometry/topology, logic and computation.
To go beyond 431 in probability you should take next Math
521 - Analysis, and after that one or both of these: Math
632 - Introduction to Stochastic Processes and Math 635
- Introduction to Brownian Motion and Stochastic Calculus.
Those who would like a proof based introduction to probability
could consider taking Math 531 - Probability Theory
(531 requires a proof based course as a prerequisite).
Probability theory is ubiquitous in natural science, social
science and engineering, so this course can be valuable in
conjunction with many different majors. Aside from being a
beautiful subject in and of itself, is used throughout the
sciences and industry. For example, in biology many models
of cellular phenomena are now modeled probabilistically as opposed
to deterministically. As for industry, many models used by
insurance and financial companies are probabilistic in nature.
Thus, those wishing to go into actuarial science or finance need
to have a solid understanding of probability. Probabilistic models
show up in the study of networks, making probability theory useful
for those interested in computer science and information
technology.
Challenge
Problems
We will also post challenge problems during the semester. These will be graded separately from the regular homework assignments. The total score from the challenge problems will be normalized at the end of the semester, usually the best problem solvers get a couple of extra points towards their final grade. These problems are not graded like the rest of the homework, with ordinary standards of partial credit. The scale for the challenge problems is 0-3, with 0 unless you make significant progress towards the answer. Note that you do not need to work on these problems if you do not want to, they are there for the students who enjoy working on harder problems.
Here is a tentative weekly schedule, to be adjusted as we go.
Weeks 1-2 Axioms of probability, sampling,
consequences of the rules of probability
Weeks 3-4 Conditional probability and
independence
Weeks 5-6 Random variables (distribution,
expectation, variance)
Weeks 6-7 Normal and Poisson approximation of
binomial distribution
Week 8 Moment generating
function and transformation of random variables
Weeks 9-12 Joint behavior of multiple random variables
(joint distribution, expectation, covariance)
Week 13 Estimating tail
probabilities, the Law of large numbers and the Central limit
theorem
Weeks 14-15 Conditional distribution and conditional expectation
The Math Club provides interesting lectures and other math-related events. Everybody is welcome.