Credits: 5

Schedule: 07.01.2020 - 17.02.2020

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

Lecturer: Prof Lasse Leskelä

Teaching assistant: MSc Joona Karjalainen

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

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): 


The course grade g is determined by normalized exam points (e = E/Emax), normalized homework points (h = H/Hmax), and normalized quiz points (q = Q/Qmax)
according to

           g = f( max( 1.00*e, 0.50*e + 0.40*h + 0.10*) )

where f: [0,1] → {0,1,2,3,4,5} is a deterministic increasing function such that f(0.5) ≥ 1 and f(0.9) ≥ 5.


There are two optional dates for taking the course exam:

  • Mon 17 Feb 2020
  • Wed 8 Apr 2020

The points from homeworks and quizzes are valid in both of the above exams, but not thereafter.

You may 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 course exam contains 4 problems each worth 6 points.

Workload (valid 01.08.2018-31.07.2020): 

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


Details on calculating the workload (applies in this implementation): 

In addition to regular exercise sessions, every lecture is preceded by an online quiz which provides additional exercises and helps you to warm up before the lecture.

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 main course material is the lectures notes:

The lectures notes contain everything that is required to complete this course. For those who prefer to read a standard textbook instead, one option is the official course textbook

  • J Jacod & P Protter. Probability Essentials. Springer 2004.
Another textbook, with essentially the same contents compressed into a smaller number of pages, is

  • D Williams. Probability with Martingales. Cambridge University Press 1991.

If you are planning to continue your studies in stochastics further, possibly to a doctoral level, a recommended option is to consider the following reference book which contains a vast amount of content in probability theory:

  • O Kallenberg. Foundations of Modern Probability. 2nd edition. Springer 2002.

Substitutes for Courses (valid 01.08.2018-31.07.2020): 



Course Homepage (valid 01.08.2018-31.07.2020):

Prerequisites (valid 01.08.2018-31.07.2020): 

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

Grading Scale (valid 01.08.2018-31.07.2020): 



Details on the schedule (applies in this implementation): 

Lectures:Mon & Wed 10–12 @ M3 (Otakaari 1) (Exception: 6 Jan 10–12 -> 7 Jan 8–10)

Exercises: Wed 14–16 @ Y307 (Otakaari 1)

The first lecture is moved to 7 Jan due to Loppiainen holiday


Registration and further information