### Course home page

##### Course description

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.##### Contents

- 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

##### Time

2016-2017 Period III (6 weeks)- Lectures: Mon 10-12 in M3 and Tue 12-14 in M2 (2 x 2h lectures / week)
- Exercise sessions: Thu 14-16 in M2 (1 x 2h exercises / week)

##### Prerequisites

Familiarity with continuous functions and open sets (e.g. MS-C1540 Euklidiset avaruudet).##### Grading

The 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

##### Exam

The exam will be held on:- Monday 13.02.2017 at 13:00-16:00

- Tuesday 04.04.2017 at 13:00-16:00

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.

##### Exercises

There are weekly problem sets, posted under the "Assignments" tab on this page. Written solutions to the problems are to be returned to the course homework folder (on the announcement board next to office Y249d) 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 Thursdays 14-16, the course teaching assistant Joona Karjalainen 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!

##### Quizzes

Before 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.##### Literature

Textbook- J. Jacod & P. Protter:
*Probability Essentials*. Universitext, Springer, 2004.

Alternative textbook

- D. Williams:
*Probability with Martingales*. Cambridge University Press, 1991.

Yet another alternative textbook in Finnish

- T. Sottinen:
*Todennäköisyysteoria*. (online lecture notes)

##### Responses to feedback

Thank you to everyone who gave feedback! Below I respond to the issues that came up most frequently, and that we could possibly affect by course arrangements.

By far the most common criticism was about the deadline for turning in the solutions to the exercises. Our chosen exercise format itself, in which solutions are to be turned in in written and the TA gives hints in the exercise session, got only positive feedback. With the format we had actually arranged more time for the students to turn in their solutions compared to the previous year. Unfortunately, our options for arranging even more time during the 6 weeks course are limited. The current schedule is already essentially as late as possible, since we are not allowed to have exercises during the exam week. As a partial solution, we may try to move the exercise session to earlier on during the week. We will also emphasize that students are expected to think about the problems even before the exercise session, where hints are given.

The online quizzes before lectures got only positive comments. Now that they have been implemented once, it will be relatively easy to include them in the future courses as well, and improve them further.

The lecture notes got both positive and negative feedback. The negative feedback pertained to the hand-written notes, which were found not the easiest to read, and which lack some search features of typeset electronic files. Their intended function was of course not to replace course textbooks, but to serve as a record of the lecture contents. We are working towards writing typeset lecture notes that cover the scope of this concise course. Fortunately, even with the current hand-written notes, the positive comments about the notes were still more numerous than critical ones.

Some responses noted the course emphasis on theoretical content, especially measure theory. We have indeed tried to be very explicit about this: the course name has been changed to "Probability theory", and the first lecture begins by discussing the relationship between probability theory and measure theory. Measure theoretic probability is necessary for advanced topics in stochastics and statistics. In the current Master's programme, we have chosen to cover the theory in this 6 weeks course in period III, which is always followed by another course in period IV applying the theory.

Overall, the feedback was for the most part positive. We try to use it to develop the course further. Thank you for all the responses!