### MS-E1600 - Probability Theory D, Lecture, 8.1.2024-19.2.2024

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.

#### LEARNING OUTCOMES

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:

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):

LecturerProf Lasse Leskelä

Teaching assistantMSc 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

##### Content
• 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

##### Assessment Methods and Criteria
• valid for whole curriculum period:

Weekly exercises and exam.

• applies in this implementation

##### Evaluation

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.

• 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 or 2 days before the exam)

• valid for whole curriculum period:

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

#### DETAILS

##### Substitutes for Courses
• valid for whole curriculum period:

##### Prerequisites
• valid for whole curriculum period:

#### FURTHER INFORMATION

##### 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 (sisu.aalto.fi).