Spring 2016
Meetings: TuTh 9:30AM  10:45AM, B139 Van Vleck
Instructor: Benedek Valkó
Grader: Hans
Chaumont
Office: 409 Van Vleck
Phone: 2632782
Email: valko at math dot wisc dot edu
Office hours: Tu 1pm2pm or by appointment
This is the course homepage that also serves as the syllabus for the course. Here you will find homework assignments, our weekly schedule, and updates on scheduling matters. The Mathematics Department also has a general information page on this course.
I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.
Math 531 is a rigorous introduction to probability theory on an advanced undergraduate level with an emphasis on problem solving. Only a minimal amount of measure theory is used (in particular, Lebesgue integrals will not be needed). The course gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some of the classical results of probability theory with proofs (DeMoivreLaplace limit theorems, the study of simple random walk on Z, applications of generating functions).
Possible subsequent probability courses are Math 632  Introduction to Stochastic Processes and Math 635  Introduction to Brownian Motion and Stochastic Calculus.
A proof based analysis course (421 or 521 or 375376) or consent
of instructor.
Course grades will be based on quizzes, home work and reading
assignments (20%), two midterm exams (20% each) and the final exam
(40%).
The midterm exams will be evening exams scheduled on Wednesdays.
Please check the dates at the beginning of the semester. There
will not be any makeup exams for the midterms.
Exam dates:
Homework and reading assignments will be posted on the Learn@UW
site of the course. The assignments will be usually due on
Thursdays at the beginning of the class.
Homework assignments will contain exercises related to the covered
material. The reading assignments will include material that is
not covered (or not covered in detail) in the class, together with
related exercises. I plan to have a mixture of homework and
reading assignments during the semester.
Covered
topics 
Suggested reading for next
week 

Week 1 (1/19, 1/21) 
Warmup:
random experiments with equally likely outcomes The birthday problem, Buffon's needle problem The Kolmogorov axioms of probability, the probability space of infinitely many coin flips simple consequences of the additivity property 
Sections
1.21.4 
Week 2 (1/26, 1/28) 
Random
variables, consequences of the additivity property,
inclusionexclusion, the continuity of probability measure, 
Sections 1.41.6 
Week 3 
BorelCantelli lemma,
conditional probability, independence of events 
Sections 2.1, 2.3, 2.4 
Week 4 
Independence of events,
constructing independent experiments, Distribution of random variables, the probability mass function, the cumulative distribution function 
Sections 2.5, 3.13.3,
4.14.3 
Week 5 
Probability density
function, joint distribution of several random variables, Joint CDF, PMF and PDF, 
Sections 3.3, 4.3 
Week 6 
Independence of random
variables, marginal distribution from joint distribution Expectation of random variables, general definition, discrete and continuous cases, properties of expectation 
Sections 3.3, 3.4, 4.3, 7.3

Week 7 
Further properties of expectation, the indicator method, variance and moments,  Section 3.10 
Week 8. 
Coupon collector's problem,
Markov and Chebyshev's inequality, Weak Law of Large numbers


Week 9 
Strong Law of Large Numbers
with finite fourth moment, Random walks 

Week 10 
The normal distribution, Stirling's formula,
De MoivreLaplace Central Limit Theorem, scaling limits of
the symmetric random walk The Poisson limit of binomial distribution. 

Week 11 
Applications of the CLT, sum of independent
random variables 
Sections 4.12, 3.7, 4.6 
Week 12. 
Law of rare events, Conditional distribution, conditional expectation  Sections 5.1, 5.7 
Week 13. 
Conditional distribution, conditional
expectation, Generating functions of random variables 
Sections 5.1, 5.3, 5.4, 5.7 
Week 14. 
Generating functions of random variables Applications: sums of independent random variables, random walks, branching processes 
Sections 5.9, 5.10 
Week 15. 
Generating functions and limits in
distribution 