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: 09.01.2023 - 20.02.2023

Teacher in charge (valid for whole curriculum period):

Teacher in charge (applies in this implementation): Lasse Leskelä

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

Lecturer: Prof Lasse Leskelä

Teaching assistant: MSc Kalle Alaluusua

CEFR level (valid for whole curriculum period):

Language of instruction and studies (applies in this implementation):

Teaching language: English. Languages of study attainment: English


  • valid for whole curriculum period:

    - 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

  • applies in this implementation


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

               g = f( max( 1.00*e, 0.60*e + 0.30*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.

Assessment Methods and Criteria
  • valid for whole curriculum period:

    Weekly exercises and exam.

  • valid for whole curriculum period:

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


Substitutes for Courses


Further Information
  • valid for whole curriculum period:

    Teaching Language : English

    Teaching Period : 2022-2023 Spring III
    2023-2024 Spring III

    Enrollment :

    Registration takes place in Sisu (