Math/Stat 431 Introduction to the Theory of Probability
Fall 2013 Lectures 1 and 3 MWF
|Meetings: MWF, Lecture 1 9:55-10:45, Lecture 2 1:20-2:10, Van Vleck B115|
|Office: 419 Van Vleck. Office Hours:
Mondays and Wednesdays after classes, other times by appointment. |
|Phone: 263-2812 |
seppalai and then at math dot
wisc dot edu (I can answer questions by email.) |
This is the course homepage that also serves as the syllabus
for the course. Here you will find
our weekly schedule and updates on scheduling matters.
The Mathematics Department also has a general
on this course.
Math 431 is an introduction to
probability theory, the
mathematics that studies random phenomena. We model simple random experiments mathematically
and learn techniques for studying these models. Topics covered
include methods of counting
(combinatorics), axioms of probability,
random variables, the most important discrete and
continuous probability distributions, expectations, moment
conditional probability and conditional expectations,
multivariate distributions, Markov's and Chebyshev's inequalities,
laws of large numbers, and the central limit theorem.
Probability theory is ubiquitous
in natural science, social science and engineering,
so this course can be valuable in conjunction with many different
majors. 431 is not a course in 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
521 Analysis, and after that one or both of these:
632 Introduction to Stochastic Processes and
635 Introduction to Brownian Motion and Stochastic Calculus.
To be technically prepared for Math 431 one needs to be
comfortable with the language of sets and calculus, including multivariable
calculus, and be ready for abstract reasoning. Probability theory can seem
very hard in the beginning, even after success in past math courses.
A First Course in Probability, Eighth Edition, by S. Ross.
Note that the 8th edition is not the newest
But other editions of Ross's book cover the subject matter also.
The only possible harm from using a different edition
is that you may have to look up the homework problems from the 8th edition.
We will use Learn@UW to post homework assignments, solutions to homework, and lecture notes. The lecture notes come from a book manuscript that some instructors are working on.
The lecture notes evolve (and hopefully improve) so if you print, do not print a large number of pages at one time.
Three books are on reserve in the library in case you wish to see how other authors develop the material.
You can also find old 431 exams in the Math Library Course Reserves. But note that these exams might not necessarily be a perfect fit for this year's course.
- Probability, by Jim Pitman. This was the course textbook in 2012-2013.
- Probability, statistics, and stochastic processes, by Peter Olofsson.
- Probability and random processes, by Geoffrey Grimmett.
Course grades will be based on homework
and quizzes (20%), three midterm exams (3x15%),
and a comprehensive final exam (35%).
Midterm exams will be in the evenings of these dates.
No calculators, cell phones, or other gadgets
will be permitted in exams and quizzes,
only pencil and paper.
- Exam 1 Wednesday October 2, 7:15-8:45 PM, Ingraham B10
- Exam 2 Wednesday October 30, 7:15-8:45 PM, Ingraham B10
- Exam 3 Wednesday November 20, 7:15-8:45 PM, Ingraham B10
Here are grade lines that can be guaranteed in advance.
A percentage score in the
indicated range guarantees at least the letter grade next to it.
- Lecture 1 final exam 12/19/2013, Thursday 12:25 PM - 2:25 PM, Soc Sci 6203.
- Lecture 3 final exam 12/18/2013, Wednesday 5:05 PM - 7:05 PM, Van Vleck B239.
A [100,89), AB [89,87), B [87,76), BC [76,74), C [74,62), D
[62,50), F [50,0].
Here is a tentative weekly schedule, to be adjusted as we go.
The numbers refer to sections
in the eighth edition of Ross.
- Week 1. (9/3-6) Axioms of probability, equally likely outcomes, infinitely many outcomes. 1.1-1.4, 2.2-2.3, 2.5. Note that Ross's Chapter 1 is all about methods of counting. We shall not reserve time for this chapter but will discuss counting as part of the topic of equally likely outcomes.
- Week 2. (9/9-13) Consequences of the axioms, random variables, probability mass function of a discrete random variable, conditional probability, Bayes' rule. 2.4, 3.2-3.3, 4.1-4.2.
- Week 3. (9/16-20) Independence, binomial and geometric distribution, birthday problem. 2.5, 3.4, 4.6, 4.8.1, 6.2.
- Week 4. (9/23-27) Random variables, density function, cumulative distribution function, expectation and variance. 4.3-4.5, 5.1-5.3.
- Week 5. (9/30-10/4) Normal distribution, normal approximation of the binomial. 5.4.
- Week 6. (10/7-11) Normal approximation of the binomial, Poisson distribution, exponential distribution, Poisson approximation of the binomial. 4.7, 5.4.1, 5.5.
- Week 7. (10/14-18) Moment generating function. Density of g(X). Catching up if necessary. 5.7, 7.7.
- Week 8. (10/21-25) Joint distributions. 6.1, 6.3.
- Week 9. (10/28-11/1) Sums of independent random variables, convolution. 6.2, 6.3.
- Week 10. (11/4-8) Convolution, symmetry and exchangeability, indicator method. 6.3, 6.8, 4.9, 7.1-7.2.
- Week 11. (11/11-15) Hypergeometric distribution. Expectations of independent products, variance of a sum of independent random variables, covariance and correlation. 4.8.3, 7.4, 7.7.
- Week 12. (11/18-22) Covariance and correlation. 7.4.
- Week 13. (11/25-27) Markov's and Chebychev's inequalities, law of large numbers, central limit theorem. 8.1-8.4 Thanksgiving break.
- Week 14. (12/2-6) Conditional distributions and expectations. 6.4-6.5, 7.5-7.6.
- Week 15. (12/9-13) Properties of normal distributions, further examples, review. 7.8.
Homework assignments will be posted on Learn@UW. Homework assignments are due Thursdays 11 AM in the instructor's office or mailbox, unless otherwise indicated.
Larger list of suggested exercises,
not for handing in
The numbers refer to exercises in the eighth edition of Ross.
- Chapter 2
Problems: 1, 3, 6, 9, 13, 18, 27, 41, 45, 47, 50.
Theoretical Exercises: 6, 10, 11.
- Chapter 3
Problems: 1, 2, 7, 10, 13, 15, 16, 17, 18, 38, 47, 53, 55, 56, 57, 74, 76, 77, 78, 79,
Theoretical Exercises: 1, 2, 5, 6, 9.
- Chapter 4
Problems: 1, 4, 8, 20, 21, 22, 23, 38, 43, 50, 51, 52, 59, 60, 61, 72, 75, 84.
Theoretical Exercises: 2, 3, 4, 5, 6, 7, 9, 11, 14, 25, 27.
- Chapter 5
Problems: 1, 3, 5, 7, 10, 13, 18, 19, 27, 28, 33, 37, 39, 40.
Theoretical Exercises: 1, 5, 11, 12, 13, 15, 30, 31.
- Chapter 6
Problems: 1, 2, 4, 6, 8, 9, 10, 12, 17, 18, 23, 28, 33, 34, 36, 37,
38, 39, 42, 48.
Theoretical Exercises: 2, 5, 6, 9, 15, 17.
- Chapter 7
Problems: 9, 16, 19, 21, 22, 26, 30, 32, 37, 45, 47, 57, 75, 76.
Theoretical Exercises: 9, 10.
- Chapter 8
Problems: 1, 4, 5, 6, 7, 11, 14, 15.
Instructions for homework
- Observe rules of academic integrity.
Handing in plagiarized
work, whether copied from a fellow student or off the web, is not acceptable.
Plagiarism cases will lead to sanctions.
No late papers will be accepted. You can bring the homework earlier to
the instructor's office or mailbox.
Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible.
If the answer is a simple fraction or expression,
a decimal answers from a calculator is not necessary. But for some
exercises you need a calculator to get the final answer.
Answers to some exercises are in the back of
the book, so answers alone carry no credit. It's all in the reasoning
you write down.
Put problems in the correct order and staple your pages together.
Do not use paper torn out of a binder.
Be neat. There should not be text crossed out.
Recopy your problems. Do not hand in your rough draft or first attempt.
Papers that are messy, disorganized or unreadable cannot be graded.
The Math Club
lectures and other math-related events.
Everybody is welcome.