Math/Stat 431 Introduction to the Theory of Probability
Fall 2014 Lecture 2 MWF
Meetings: MWF 8:509:40 Van Vleck B139 
Instructor:
Timo Seppäläinen

Office: 425 Van Vleck. Office Hours:
Mondays and Wednesdays 1112, other times by appointment. 
Phone: 2633624 
Email:
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
information page
on this course.
Course description
Math 431 is an introduction to
probability theory, the
part of
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
generating functions,
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.
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.
Prerequisites
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.
Textbook
The course follows lecture notes authored by David Anderson, Timo Seppäläinen, and Benedek Valkó. These will be provided to the students at no cost.
Learn@UW
We use Learn@UW to post homework assignments, solutions to homework, and lecture notes.
The lecture notes evolve (and hopefully improve) so if you print, do not print a large number of pages at one time.
Library reserves
Three books are on reserve in the library in case you wish to see how other authors develop the material.
 A First Course in Probability, by Sheldon Ross.
 Probability, by Jim Pitman.
 Probability, statistics, and stochastic processes, by Peter Olofsson.
 Probability and random processes, by Geoffrey Grimmett.
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.
Evaluation
Course grades will be based on homework
and quizzes (15%), three midterm exams (3x15%),
and a comprehensive final exam (40%). No calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencil and paper. Midterm exams will be in class on the following dates:
 Exam 1 Wednesday October 1
 Exam 2 Wednesday October 22
 Exam 3 Wednesday November 19
 Final exam 12/17/2014, Wednesday 7:259:25 PM, room to be announced.
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.
[100,89) A, [89,87) AB, [87,76) B, [76,74) BC, [74,62) C, [62,50) D, [50,0] F.
Fall 2014 daily schedule
Here we record the sections of the notes as we go through them.
Week  Monday  Wednesday  Friday 
1 9/29/5
 
1.2 Axioms of probability.  1.3 Sampling. 
2 9/89/12 
1.4 Infinitely many outcomes. Review of geometric series. 1.5 Inclusionexclusion rules. 
1.5 Using inclusionexclusion and complements to calculate probabilities. DeMorgan's laws.  1.6 Random variables. Probability mass function of a discrete random variable. 
3 9/1519 
2.1 Conditional probability. 2.2 Bayes formula. 
2.2 Bayes formula. 2.3 Independence. Quiz on Chapter 1.  2.1 Conditional probability obeys the rules of probability. 2.3 Independence of 3 or more events. Independence of random variables. 2.4 Independent trials, Bernoulli and binomial distribution. 
4 9/2226 
2.4 Independent trials, geometric distribution. 2.5 Conditional independence.  2.5 Birthday problem. 3.1 Probability mass functions, probability density functions. Quiz on Chapter 2.  3.1 Probability mass functions, probability density functions, cumulative distribution functions. 
5 9/2910/3 
Review of sampling, this time in terms of random variables. 3.2 Expectation EX for discrete and continuous random variables.  Exam 1  3.2 Expectation E[g(X)] and variance Var(X) for discrete and continuous random variables. 
6 10/610  3.2 Expectation and variance of aX+b. 3.3 Normal distribution. 
4.1 Normal approximation of the binomial. 
4.1 Normal approximation of the binomial with continuity correction. 
7 10/1317 
4.1 Law of large numbers for binomial. Estimating an unknown success probability. 
4.1 Estimating an unknown success probability. Confidence intervals and polling.  4.2 Poisson distribution and the Poisson approximation of the binomial. 
8 10/2024 
4.2 Poisson approximation of the binomial. 4.3 Exponential distribution. 
Exam 2  5.1 Moment generating function. 
9 10/2731 

 
10 11/37 

 
11 11/1014 

 
12 11/1721 

Exam 3  
13 11/2426 

 Thanksgiving 
14 12/15 

 
15 12/812 

 
Tentative weekly schedule for fall 2014
Here is a tentative weekly schedule that looks into the future, to be adjusted as we go. Section numbers refer to the lecture notes.
 Week 1. (9/25) Axioms of probability, equally likely outcomes. 1.11.3.
 Week 2. (9/812) Infinitely many outcomes, consequences of the axioms, random variables, probability mass function of a discrete random variable. 1.41.6.
 Week 3. (9/1519) Conditional probability, Bayes' rule, independence, binomial and geometric distribution, birthday problem. 2.12.5.
 Week 4. (9/2226) Random variables, density function, cumulative distribution function. 3.1.
 Week 5. (9/2910/3) Expectation and variance. 3.2.
 Week 6. (10/610) Normal distribution, normal approximation of the binomial, Poisson distribution, exponential distribution, Poisson approximation of the binomial. 3.3, 4.14.4.
 Week 7. (10/1317) Moment generating function. Density of g(X). 5.15.2.
 Week 8. (10/2024) Joint distributions. 6.1, 6.2.
 Week 9. (10/2710/31) Sums of independent random variables, convolution, negative binomial distribution, Poisson process. 6.3.
 Week 10. (11/37) Exchangeability, hypergeometric distribution, indicator method for expectations. 6.5, 7.1.
 Week 11. (11/1014) Expectations of independent products, variance of a sum of independent random variables, covariance and correlation. 7.27.4.
 Week 12. (11/1721) Covariance and correlation. 7.4.
 Week 13. (11/2426) Markov's and Chebychev's inequalities, law of large numbers, central limit theorem. 8.18.3. Thanksgiving break.
 Week 14. (12/15) Conditional distributions and expectations. 9.19.3.
 Week 15. (12/812) Remaining topics, examples, review.
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
provides interesting
lectures and other mathrelated events.
Everybody is welcome.