Vai al contenuto principale Vai al footer
Università degli Studi di Parma Università degli Studi di Parma
Degree course in
Mathematics
Università degli Studi di Parma
Department of Mathematical, Physical and Computer Sciences

ELEMENTS OF PROBABILITY
cod. 13473

Academic year 2015/16
3° year of course - Second semester
Professor
Domenico MUCCI
Academic discipline
Probabilità e statistica matematica (MAT/06)
Field
Formazione modellistico-applicativa
Type of training activity
Characterising
72 hours
of face-to-face activities
9 credits
hub: PARMA
course unit
in - - -
lectures timetable unavailable
exam calls
Print

Learning objectives

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

Prerequisites

- - -

Course unit content

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.

Full programme

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.

Bibliography

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

Teaching methods

Lectures

Assessment methods and criteria

Written and oral examination

Other information

- - -

2030 agenda goals for sustainable development

- - -

Contacts

Toll-free number

800 904 084

Student registry office

E. segreteria.scienze@unipr.it
T. +39 0521 905116

Quality assurance office

Education manager
dott.ssa Giulia Bonamartini
T. +39 0521 906968
E. servizio smfi.didattica@unipr.it
E. del manager giulia.bonamartini@unipr.it

President of the degree course

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Faculty advisor

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Career guidance delegate

Prof. Francesco Morandin
E. francesco.morandin@unipr.it

Tutor Professors

Prof. Emilio Acerbi
E. emilio.acerbi@unipr.it

Prof. Marino Belloni
E. marino.belloni@unipr.it

Prof.ssa Maria Groppi
E. maria.groppi@unipr.it

Prof.ssa Chiara Guardasoni
E. chiara.guardasoni@unipr.it

Prof. Luca Lorenzi
E. luca.lorenzi@unipr.it

Prof. Costantino Medori
E. costantino.medori@unipr.it

Prof. Adriano Tomassini
E. adriano.tomassini@unipr.it

Erasmus delegates

Prof.ssa Fiorenza Morini
E. fiorenza.morini@unipr.it

Quality assurance manager

Prof.ssa Maria Groppi
E. maria.groppi@unipr.it

Tutor students

Dott. Matteo Mezzadri
E. matteo.mezzadri@studenti.unipr.it