This course is about the mathematical foundations of randomness. Most advanced topics in stochastics and statistics rely on probability theory. The basic constructions are identical to measure theory, but there are a number of distinctly probabilistic features such as independence, notions of convergence of random variables, information contained in a sigma-algebra, conditional expectation, characteristic functions and generating functions, laws of large numbers and central limit theorems, etc.
- Random numbers, vectors, and sequences
- Integration with respect to a probability measure
- Stochastic independence and product measure
- Law of large numbers and the central limit theorem
- (Conditional expectation with respect to a sigma-algebra)
Time2018-2019 Period III (6 weeks)
- Lectures: Mondays 10-12 in M3 and Wednesdays 10-12 in M3 (2 x 2h lectures / week)
- Exercise sessions: Wednesdays 14-16 in Y307 (1 x 2h exercises / week)
PrerequisitesFamiliarity with continuous functions and open sets (e.g. MS-C1540 Euklidiset avaruudet).
GradingThe course grade is determined by whichever of the following scores is higher
- 100% exam score
- 50% exam score + 10% score of quizzes + 40% homework solutions
ExamThe exam will be held on:
- Monday 18.02.2019
- Wednesday 10.04.2019
You are allowed to bring to the exam a handwritten memory aid sheet. The memory aid sheet must be of size A4 with text only on one side, and it must contain your name and student number in the upper right corner. You don’t need to return your memory aid sheet. The exam consists of 4 problems, each worth 6 points.
ExercisesThere are weekly problem sets, posted under the "Assignments" tab on this page. Written solutions to the problems are to be turned in to the course homework "mailbox" (outside the Laskutupa room Y190c) or electronically through MyCourses (see Assignments) by Mondays at 10 am. These solutions amount to 40% of the course grade, except if the exam score alone is higher.
In the exercise sessions on Wednesdays 14-16, the course teaching assistant Niko Lietzén will provide help in solving the problems. The exercise sessions also contain brief recitations on the topics of the lectures and problems. You should think about the problems in advance, so as to be able to focus on whatever you find difficult when the teaching assistant is there to instruct you!
QuizzesBefore each lecture, you are expected to answer simple quizzes about preliminary material posted under the "Lectures" tab. The purpose is to make sure that you are familiar with the basic concepts needed to follow the lecture. These quizzes amount to 10% of the course grade, except if the exam score alone is higher.
LiteratureThe course primarily follows lecture notes provided in the Materials section. Many textbooks have similar content, e.g.,
- J. Jacod & P. Protter: Probability Essentials. Universitext, Springer, 2004.
- D. Williams: Probability with Martingales. Cambridge University Press, 1991.
- R. Durrett: Probability: Theory and Examples. Cambridge University Press, 2010.
- T. Sottinen: Todennäköisyysteoria. (online lecture notes)