### MS-E1600 - Probability theory, 07.01.2019-18.02.2019

Credits: 5

Schedule: 07.01.2019 - 18.02.2019

Contact information for the course (applies in this implementation):

Lecturer: Kalle Kytölä

Teaching assistant: Niko Lietzén

Teaching Period (valid 01.08.2018-31.07.2020):

III Spring (2018-2019, 2019-2020)

Learning Outcomes (valid 01.08.2018-31.07.2020):

After completing the course, the participant

- Can compute the expected value of a random number as an integral with respect to a probability measure
- Can compute probabilities related to independent random variables by using a product measure
- Recognizes different types of convergence of a random sequence
- Can explain how and when a random sum can be approximated by a Gaussian distribution
- Can represent conditional probabilities with respect to the information content of a sigma-algebra

Content (valid 01.08.2018-31.07.2020):

- Random numbers, vectors, and sequences
- Describing information using sigma-algebras
- Integration with respect to a probability measure
- Stochastic independence and product measure
- Law of large numbers and the central limit theorem

Details on the course content (applies in this implementation):

##### Course 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)

Assessment Methods and Criteria (valid 01.08.2018-31.07.2020):

Weekly exercises and exam.

Elaboration of the evaluation criteria and methods, and acquainting students with the evaluation (applies in this implementation):

Exercises: There are 6 problem sets during the course, one each week. Written solutions are turned in for grading.

Quizzes: Before each lecture there is an online quiz in MyCourses.

Exam: The exam consists of 4 problems, to be completed in 3 hours.

Grade: 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

2 x 2h lectures, 1 x 2h exercises sessions

Study Material (valid 01.08.2018-31.07.2020):

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

Details on the course materials (applies in this implementation):

The course follows lecture notes provided on the MyCourses page.

Substitutes for Courses (valid 01.08.2018-31.07.2020):

Mat-1.3601

Course Homepage (valid 01.08.2018-31.07.2020):

https://mycourses.aalto.fi/course/search.php?search=MS-E1600

Prerequisites (valid 01.08.2018-31.07.2020):

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