Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.


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

Credits: 5

Schedule: 11.01.2021 - 22.02.2021

Teacher in charge (valid 01.08.2020-31.07.2022): Lasse Leskelä

Teacher in charge (applies in this implementation):

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

CEFR level (applies in this implementation):

Language of instruction and studies (valid 01.08.2020-31.07.2022):

Teaching language: English

Languages of study attainment: English


  • Valid 01.08.2020-31.07.2022:

    - 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 01.08.2020-31.07.2022:

    Weekly exercises and exam.

  • Applies in this implementation:


    The course grade g is determined by normalized exam points (= E/Emax), normalized homework points (= H/Hmax), and normalized quiz points (= 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.

  • Valid 01.08.2020-31.07.2022:

    2 x 2h lectures, 1 x 2h exercise sessions with weekly homeworks


Study Material
  • Valid 01.08.2020-31.07.2022:

    K Kytölä. Probability theory. Lecture notes. Aalto University 2019.


  • Applies in this implementation:

    Extensive stochastics "bible":

    • Olav Kallenberg: Foundations of Modern Probability. Springer 2002.

    Finnish reading material:

Substitutes for Courses
  • Valid 01.08.2020-31.07.2022:



  • Valid 01.08.2020-31.07.2022:

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