Meetings: TuTh 9:30AM - 10:45AM, B139 Van Vleck
Instructor: Benedek Valkó
Grader: Hans Chaumont
Office: 409 Van Vleck
Email: valko at math dot wisc dot edu
Office hours: Tu 1pm-2pm 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 (DeMoivre-Laplace 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 375-376) or consent
Course grades will be based on quizzes, home work and reading
assignments (20%), two midterm exams (20% each) and the final exam
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.
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.
||Suggested reading for next
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
variables, consequences of the additivity property,
inclusion-exclusion, the continuity of probability measure,
conditional probability, independence of events
||Sections 2.1, 2.3, 2.4
||Independence of events,
constructing independent experiments,
Distribution of random variables, the probability mass function, the cumulative distribution function
|Sections 2.5, 3.1-3.3,
function, joint distribution of several random variables,
Joint CDF, PMF and PDF,
|Sections 3.3, 4.3
||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
||Further properties of expectation, the indicator method, variance and moments,||Section 3.10
||Coupon collector's problem,
Markov and Chebyshev's inequality, Weak Law of Large numbers
||Strong Law of Large Numbers
with finite fourth moment, Random walks
||The normal distribution, Stirling's formula,
De Moivre-Laplace Central Limit Theorem, scaling limits of
the symmetric random walk
The Poisson limit of binomial distribution.
||Applications of the CLT, sum of independent
||Sections 4.12, 3.7, 4.6
||Law of rare events, Conditional distribution, conditional expectation||Sections 5.1, 5.7
|| Conditional distribution, conditional
Generating functions of random variables
|Sections 5.1, 5.3, 5.4, 5.7|
||Generating functions of random variables
Applications: sums of independent random variables, random walks, branching processes
|Sections 5.9, 5.10
||Generating functions and limits in