Kurssiesite

LEARNING OUTCOMES

This course equips you with a deep and rigorous understanding of probability theory. It provides the mathematical tools and abstract concepts necessary to model and analyse complex probabilistic phenomena, apply advanced statistical techniques, and engage in research and applications in mathematics, statistics, machine learning, and other fields.  What you will learn:

• Develop a strong understanding of probability spaces and sigma-algebras, enabling you to operate with random numbers, vectors, and sequences as measurable functions.
• Compute probabilities, expected values, variances, and correlations by integrating with respect to a probability measure.
• Apply products of probability measures and probability kernels to model and analyse stochastic independence and conditional probabilities.
• Get introduced to key convergence concepts of random sequences (almost sure, in probability) and classical limit laws of random sequences.
• Become familiar with key statistical distances (total variation, Wasserstein) and information-theoretic divergences (eg Kullback–Leibler) to quantify the difference between two probability distributions.

Credits: 5

Schedule: 02.09.2024 - 14.10.2024

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 (office hours Tue 12–13)

CEFR level (valid for whole curriculum period):

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

Teaching language: English. Languages of study attainment: English

CONTENT, ASSESSMENT AND WORKLOAD

Content

• valid for whole curriculum period:

  This course provides a unified foundation of probability theory in a general measure-theoretic framework where random variables are represented as functions on an abstract measurable space that is equipped with a probability measure from which all probabilities, expected values, and correlations associated with a given stochastic model can be computed. Key concepts:

  • Random numbers, vectors, and sequences as measurable functions.
  • Integration with respect to a general probability measure.
  • Probability kernels for modelling conditional probabilities.
  • Convergence of random sequences and probability measures.
  • Distances and divergences between probability measures.

Assessment Methods and Criteria

• valid for whole curriculum period:

  Exam and homeworks.

• applies in this implementation

  Evaluation

  The course grade g is determined by normalised exam points (e = E/E_max), normalised homework points (h = H/H_max), and normalised quiz points (q = Q/Q_max) according to 

             g = f( max( 1.00*e, 0.60*e + 0.30*h + 0.10*q ) )

  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, and E_max, H_max, Q_max indicate the maximum attainable points from the exam, homeworks, and quizzes.

  See http://math.aalto.fi/exams for 

  • Individual study arrangements for exams (deadline to submit the form is 1 week before the exam, but the earlier the better) 
    
  • Exam rooms (announced 1–2 days before the exam)
  
  

Workload

• valid for whole curriculum period:

  Participation in contact teaching (2 x 2h weekly lectures, 1 x 2h weekly exercise classes) and independent study.

DETAILS

Study Material

• valid for whole curriculum period:

  The main study material consists of lecture notes and homework problems delivered during the course.  Most of the material is also available in textbooks:

  • J Jacod & P Protter. Probability Essentials. Springer 2004.
  • D Williams. Probability with Martingales. Cambridge University Press 1991.
  • AN Shiryaev. Probability. Springer 1996.

  The course material and much more is also available in the reference book:

  • O Kallenberg. Foundations of Modern Probability. Springer 2021.

   

Substitutes for Courses

• valid for whole curriculum period:

  Substitutes for Courses

Prerequisites

• valid for whole curriculum period:

  Prerequisites

FURTHER INFORMATION

Further Information

• valid for whole curriculum period:

  Teaching Language: English

  Teaching Period: 2024-2025 Autumn I
  2025-2026 Autumn I

Details on the schedule

• applies in this implementation

  Weekly schedule

  When		What	Where
  Mon	10:00	Quiz DL	Online
  Mon	10–12	Lecture	Hall 216, Otakaari 4 (Konetekniikka 1)
  Tue	12–13	TA office hour	Room Y249d, Otakaari 1
  Wed	10:00	Quiz DL	Online
  Wed	10–12	Lecture	Hall 213a, Otakaari 4 (Konetekniikka 1)
  Thu	12:00	Homework DL	In class or online
  Thu	12–14	Exercise session	Hall U3, Otakaari 1

  
  Exercise sessions
  

  You will present and discuss your solutions to weekly assignments during the corresponding week's exercise session. At the start of each session, you will mark what solutions you are ready to present and the TA will ask a random person to present one problem. You will receive 2 points for each exercise you are ready to present. You can present your solutions in English or Finnish.
  

  Alternatively, you may submit your solutions in writing to the TA (for example, if you cannot attend an exercise session). Homework submissions are delivered electronically as PDFs (1 file/submission) using the links below. If you send photos, please pay attention to the readability of your answers! Unreadable answers receive no points. Each problem is graded using the scale {0,1,2}. Grade 0 implies that the solution is mostly wrong or incomplete, grade 1 implies some mistakes or somewhat incomplete despite good effort. Finally, 2 implies a (nearly) perfect solution. It is good effort and hard work that matters most!

  It is possible to obtain the full mark for the course by only taking the final exam.

  TA office hours

  TA office hours weekly on Tue 12–13 @ y249d, Otakaari 1

  Here you can meet the teaching assistant, discuss homework problems, and ask for hints.

  
  

Opintojakson kuvaus