To acquire a good knowledge of the main aspects of discrete probability, statistics and their applications.
Random variables, probability distributions, elementary stochastic processes and some limit theorems. Estimators and predictions, inference, causality and correlation. Pedagogical aspects and applications to the real world through models as percolation, random cluster model, Ising model, Markov chain Monte Carlo.
Random variables, probability distributions, elementary stochastic processes and some limit theorems. Estimators and predictions, inference, causality and correlation. Pedagogical aspects and applications to the real world through models as percolation, random cluster model, Ising model, Markov chain Monte Carlo.
scheda docente
materiale didattico
Mathematical models.
Difference equations. Equilibria, stability.
Logistic map, bifurcations. Cycles. Examples.
Statistical Mechanics models: Ising model, percolation and random cluster model.
Curie-Weiss model and metastability.
Markov Chain Monte Carlo.
S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction.
O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52
Programma
Part IIMathematical models.
Difference equations. Equilibria, stability.
Logistic map, bifurcations. Cycles. Examples.
Statistical Mechanics models: Ising model, percolation and random cluster model.
Curie-Weiss model and metastability.
Markov Chain Monte Carlo.
Testi Adottati
S.Elaydi: An introduction to difference equations - SpringerS.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction.
O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52
Bibliografia Di Riferimento
Dropbox on lineModalità Erogazione
In-person lessons (distance learning is possible on Teams)Modalità Valutazione
Written and oral exam. The written test consists of exercises similar to those carried out during the lessons. In the oral exam the student can present a chosen theme among those addressed in the course.
scheda docente
materiale didattico
2) Elements of Statistics: random sampling, definition of statistical model and statistics, sufficient/minimal/complete statistics, moment method, maximal likelihood estimators, confidence interval, hypothesis testing, examples.
3) Analysis of specific models.
- Recommended exercises on the Team of the course
- Other written material can be found on the Team of the course
Programma
1) Elements of basic probability: combinatorics, axioms of probability, conditional probability and independence, random variables (discrete and continuous) with main distributions, limit theorems, examples.2) Elements of Statistics: random sampling, definition of statistical model and statistics, sufficient/minimal/complete statistics, moment method, maximal likelihood estimators, confidence interval, hypothesis testing, examples.
3) Analysis of specific models.
Testi Adottati
- Calcolo delle probabilita' (Sheldon Ross)- Recommended exercises on the Team of the course
- Other written material can be found on the Team of the course
Bibliografia Di Riferimento
- Calcolo delle probabilita' (Sheldon Ross) - Recommended exercises on the Team of the course - Other written material can be found on the Team of the courseModalità Erogazione
Preferably in presenceModalità Frequenza
Preferably in presenceModalità Valutazione
The written part consists of exercises (time needed: 2 hours), whereas the oral part will consist of a presentation of a topic chosen by the student and other questions about theoretical results seen in class.