| Meetings: TuTh 9:30AM-10:45AM Van Vleck B123 |
| Instructor: Benedek Valkó |
| Office: 409 Van Vleck. |
| Office Hours: W 3:30PM-4:30PM or by appointment |
| E-mail: valko at math dot wisc dot edu |
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.
There will be no more quizzes in the semester.
Homework assignments will be posted on the Learn@UW site of the
course. Weekly homework assignments are usually due on Thursday
at the beginning of the class. You can also submit your solution
in an electronic form via Learn@UW, but it has to arrive by 9:30AM
on the due date. Note that there is a (short) homework assignment
due on the first Thursday (September 8).
Some homework assignments will contain bonus problems for those
who would like extra challenge. The points from the bonus problems
will be converted into extra credit at the end of the semester.
Here is a tentative weekly schedule, to be adjusted as we go.
The numbers refer to sections in lecture notes that can be found
at the Learn@UW website.
| Covered
topics |
Suggested
reading for next week |
|
| Week
1. 9/6, 9/8 |
Axioms of probability, sampling,
review of counting, infinitely many outcomes, review
of the geometric series (Sections 1.1–1.3).
|
Sections 1.4–1.5, 2.1 |
| Week
2. 9/13, 9/15 |
Rules of probability, random variables (Sections 1.4–1.5, 2.1). | Sections 2.2–2.4 |
| Week
3. 9/20, 9/22 |
Conditional probability,
Bayes formula, independence (Sections 2.1–2.3). |
Sections 2.4–2.5, 3.1 |
| Week
4. 9/27, 9/29 |
Independent trials, birthday problem (Sections 2.4–2.5). | Sections 3.1–3.2 |
| Week
5. 10/4, 10/6 |
Probability distribution of a random variable, Expectation and variance (Sections 3.1–3.2). | Sections 3.3, 4.1 |
| Week
6. 10/11, 10/13 |
Gaussian distribution, Normal approximation for the binomial distribution (Sections 4.1). Midterm 1 | Sections 4.1–4.3 |
| Week
7. 10/18, 10/20 |
Normal approximation and
the law of large numbers, confidence intervals, the Poisson
distribution. (Sections 4.1–4.3) |
Sections 4.3, 5.1 |
| Week
8. 10/25, 10/27 |
Poisson approximation,
Exponential distribution Moment generating function, using the MGF to compute moments (Sections 4.3, 5.1). |
Sections 6.1–6.3 |
| Week
9. 11/1, 11/3 |
Using the MGF to identify
the distribution, the distribution of functions of random
variables Joint distribution of discrete random variables, the joint pmf (Sections 5.1-5.2, 6.1 ). |
Sections 7.1–7.2 |
| Week
10. 11/8, 11/10 |
Joint distributions
of continuous random variables, Sums of independent random variables, (Sections 6.1-6.3, 7.1). |
Sections 8.1–8.2 |
| Week
11. 11/15, 11/17 |
Sums of independent random
variables cont., Exchangeability Expectations of sums and products (Sections 7.1-7.2, 8.1-8.2). |
Section 8.2-8.3 |
| Week
12. 11/22 |
Expectation and variance of
the sample mean, coupon collector, using MGF to compute convolution (Sections 8.2-8.3) |
9.1–9.3 |
| Week
13. 11/29, 12/1 |
Covariance and correlation
(Section 8.4). Markov’s
and Chebyshev’s inequalities, Law of large numbers, central limit theorem (9.1–9.3). Midterm 2 |
Sections 10.1–10.3 |
| Week
14. 12/6, 12/8 |
Conditional distributions (Sections
10.1–10.3).
|
Sections 10.1–10.3 |
| Week
15. 12/13, 12/15 |
Conditional distributions, review (Sections 10.1–10.3). |
The Math Club provides interesting lectures and other math-related events. Everybody is welcome.