# ELEMENTS OF PROBABILITY

## Learning outcomes of the course unit

Knoweledgs and skills: at the end of the class the student should have gained skills related to measure theory and probability theory.

The exercises aim at giving the student the ability to apply the theorethical instruments in measure theory and probability.

The student should be able t check the coherence and correctedness of results not necessarily obtained by himself.

The student will be able to communicate in a clear and precise way the mathematical concepts related to the program of the class, both from a theorethical and an applied viewpoint. The interation with the teacher should provide the student with an appropriate scientific language.

## 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, Berlino 2000.

D. Williams, Probability with martingales, Cambridge mathematical textbook, Cambridge University Press 1991.

## Teaching methods

Lectures and classroom exercises.

During the lectures the main results from measure theory and from probability theory are discussed and for almost all of them complete proofs are provided. Some examples of applications of such results are provided. The classroom exercises are aimed at showing with more details how and where the abstract results can be applied to make students understand better the relevance of what they are studying.

## Assessment methods and criteria

The exam consists of two parts: a written part and an oral part. The written part is based on some exercises and it is aimed at evaluating the skills of the student in applying the abstract results proposed during the course to some concrete situations. The oral part is aimed at evaluating the knowledge of the abstract results seen during the course and their proofs.