# ELEMENTS OF PROBABILITY

## Learning outcomes of the course unit

The aim of the course consists in providing students with the basic knowledges of Probability theory and Measure theory.

## Course contents summary

1. Some elements of combinatorics.

2. Axioms of probability.

3. Conditional probability and independence.

4. Probability on a countable space.

5. Some topics from measure theory.

Exterior measures. Construction of a measure. Caratheodory theorem. Lebesgue measure. Main properties of positive measures. Measurable functions/randon variables. Integrable functions. Monotone convergence theorem, Fatou's lemma, dominated convergence theorem. L^p spaces. L^2 viewed as an Hilbert space.

6. Independent random variables (r.v.).

7. Probability distribution on R.

8. Probability distributions on R^n.

9. Characteristic functions and their properties.

10. Sums of independent random variables.

11. Gaussian r.v.

12. Convergence of r.v. (convergence in probability, weak convergence)

13. The law of large numbers.

14. The central limit theorem.

15. Conditional expectation.

16 Martingales, sub- and supermartingales.

## Course contents

1. Some elements of combinatorics.

2. Axioms of probability.

3. Conditional probability and independence.

4. Probability on a countable space.

5. Some topics from measure theory.

Exterior measures. Construction of a measure. Caratheodory theorem. Lebesgue measure. Main properties of positive measures. Measurable functions/randon variables. Integrable functions. Monotone convergence theorem, Fatou's lemma, dominated convergence theorem. L^p spaces. L^2 viewed as an Hilbert space.

6. Independent random variables (r.v.).

7. Probability distribution on R.

8. Probability distributions on R^n.

9. Characteristic functions and their properties.

10. Sums of independent random variables.

11. Gaussian r.v.

12. Convergence of r.v. (convergence in probability, weak convergence)

13. The law of large numbers.

14. The central limit theorem.

15. Conditional expectation.

16 Martingales, sub- and supermartingales.

## Recommended readings

J. Jacob, P. Protter: Probability essentials. Springer-Verlag, Berlin 2000.

## Teaching methods

Lectures

## Assessment methods and criteria

Written and oral examination