# MATA280 Foundations of stochastics (5 cr)

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## Description

The distribution of a discrete random variable, probability generating function and moments. Random vectors, independence, and the product measure of countable spaces. Markov and Chebyshev inequalities, stochastic convergence of random variables, and weak law of large numbers. The most common discrete probability distributions.

## Learning outcomes

The course introduces the basic concepts of probability theory and discrete random variables. During the course, the student develops his or her analytical reasoning skills and, in addition, learns to share the results of his or her own reasoning with others.

After completing the course, the student

- identifies the most common discrete probability distributions,
- knows how to use the probability generating function to calculate descriptive statistics of random variables,
- knows what is the joint distribution and the boundary distribution of a random vector and is able to determine if the components of the random vector are independent,
- can explain how and when the sum of random variables can be estimated using its expected value,
- has learned how to simulate the paths of a simple random process on a computer.

## Additional information

28h lectures, 7 exercise sessions

## Description of prerequisites

Basics of sequences, series and power series (for example MATA171 Introduction to mathematical analysis 1 and MATP213 Calculus 3).

## Study materials

- Meester: A natural introduction to probability theory, chapters 1-2 and 4.1.
- Ross: A first course in probability, chapters 1-4, 6.1-6.4, 7.1-7.6 and 8.1-8.2 for discrete random variables.

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### Method 2

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### Teaching (5 cr)

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Lecturs 28h (in Finnish), 7 sets of exercises, possibly a practice work

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Lecture notes: Foundations of Stochastics

Meester: A natural introduction to probability theory, luvut 1-2 ja 4.1

Ross: A first course in probability, luvut 1-4, 6.1-6.4, 7.1-7.6 ja 8.1-8.2 diskreettien satunnaismuuttujien osalta.

**Literature:**

- Meester, R.: A natural introduction to probability theory, luvut 1-2 ja 4.1

#### Teaching

##### 10/24–12/16/2022 Lectures

##### 12/15–12/15/2022 Exam

##### 1/11–1/11/2023 Exam

### Exam (5 cr)

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Final exam