Una descrizione articolata del corso con l'illustrazione aggiornata per l'anno accademico 2018/2019 del programma, i testi consigliati, le modalità d'esame etc. è disponibile in un documento .pdf scaricabile al seguente link https://goo.gl/pRtNjc

 

Il corso di Inferenza Statistica riveste un ruolo fondamentale nell'ambito del percorso formativo di una laurea in Statistica (classe di laurea L-41).

ll corso richiede che lo studente abbia familiarità con le regole del calcolo delle probabilità e abbia dimestichezza con le nozioni di variabile aleatoria e vettore aleatorio e con le principali distribuzioni.

Nel corso viene introdotto il modello statistico probabilistico per l'osservazione di un campione casuale. Attraverso la formalizzazione del modello vengono illustrati i principi e le tecniche dell'inferenza classica (impostazione frequentista) utili per affrontare i principali problemi inferenziali: la stima puntuale e d'insieme e la verifica delle ipotesi.

Nella parte finale del corso si fornirà una breve introduzione all'impostazione inferenziale bayesiana.

Per gli studenti del corso di laurea in Statistica Gestionale è parte integrante del corso il laboratorio di inferenza. Il laboratorio consiste in 12 sessioni di introduzione all'utilizzo del software statistico R (gratuito e con licenza GPL2, open source) per una migliore comprensione degli strumenti teorici e pratici dell'inferenza. Ciascuna sessione si svolge in aula informatica. Lo studente apprende l'utilizzo del software replicando le istruzioni predisposte in un'attività guidata e le rielabora durante la settimana per risolvere i quesiti finali proposti.

All'inizio di ciascun laboratorio potranno essere programmati dei brevi quiz di verifica dell'apprendimento.

Corso dedicato allo svolgimento degli esami

Obiettivo formativo principale del Laboratorio è l’apprendimento delle attività da svolgere in corrispondenza di ciascuna delle fasi in cui si articola il disegno di una ricerca sociale, dalla sua progettazione e ideazione fino alla rilevazione sul campo, all’analisi dei dati e alla loro rappresentazione all’interno di un report statistico. A tal fine le lezioni frontali e le attività pratiche previste nel programma si alternano per consentire gli studenti di mettere in pratica quanto di volta in volta illustrato dalla docente.

Learning goals.

General learning targets: 

The course is organised as a series of classes where the students will have the possibility to solve and discuss the solutions of a series of advanced exercises on the theory of stochastic processes.  

Knowledge and understanding.

At the end of the course the students will acquire the ability to solve autonomously simple and more advanced exercises on the theory of stochastic processes.  

Applying knowledge and understanding.

During the course the students will exercise the ability to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space and to solve simple applied problems in new environments and broader contexts. 

Making judgements At the end of the course the students will have the tools to evaluate critically and choose between different stochastic models to model phenomena that evolve in time and space.  

Communication skills.

The students will exercise the intuition and the communication skills necessary to describe phenomena in the mathematical language of stochastic processes. 

In particular the student will acquire familiarity with the main ideas that are behind the stochastic model, e.g., the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour. Learning skills.

The students will acquire autonomy in studying more advanced theoretical aspects of stochastic processes and in applying the main ideas of stochastic processes to the subsequent studies in the area of statistics and finance.

Prerequisites:

- elementary probability theory,

- analysis (measure theory and linear functional analysis). 

- ordinary and partial differential equations,

- linear algebra.  

Random walks (about 6 hours)

Exercises on: Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions.    

Brownian motion (about 6 hours)  

Exercises on: path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. 

Branching processes (about 2 hours)

Exercises on: expectation and variance of the population size, probability of extinction of the population.   

   

Markov chains  (about 6 hours)

Exercises on: transition matrix, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states.  

Poisson processes (about 2 hours) 

Exercises on the main properties of Poisson processes.  

Stationary processes (about 2 hours)

Exercises on: variance and covariance function, linear predictions, spectral theorem for autocorrelation functions, ergodic theorem.    

The course is organised in a series of taught classes where the students will have the possibility to solve and discuss the solutions of a series of simple and more advanced exercises on the theory of stochastic processes.   

The students are required to prepare a short written essay where they solve 4 exercises on the topics studied during the course. 

In the solution of the exercises the students must show that they have acquired the techniques to solve the problems, the intuition, the technical language, and that they are able to present the topics with all necessary details.   

Lingua inglese, Dipartimento di Scienze Statistiche

The course provides a broad introduction to stochastic processes. 

In particular the aim is  

- to give a rigorous introduction to the theory of stochastic processes, 

- to discuss the most important stochastic processes in some depth with examples and applications, 

- to give the flavour of more advanced work and applications,    

- to apply these ideas to answer basic questions in several applied situations including biology, finance and search engine algorithms.

At the end of the course the students 

- will be familiar with the basic concepts of the theory of stochastic processes in discrete and continuous time and will be able to apply various techniques to study stochastic models that appear in applications,

- will have the tools to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space and to solve simple applied problems in new environments and broader contexts,

- will have the tools to evaluate critically and choose between different stochastic models to model phenomena that evolve in time and space, 

- will acquire the necessary language skills to read academic books on the topic and research papers,

- will acquire the rationale behind the stochastic model studied (e.g. the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour...) that is necessary to communicate to specialist and non-specialist audiences.     

- will acquire the methodology and the language to study in a manner that may be largely autonomous and to apply the methodology to the subsequent studies in the area of statistics and finance.

Program:  

Random walks (about 18 hours)

Definition, Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions, short introduction on Black–Scholes model.    

Brownian motion (about 18 hours)  

Definition and existence, Brownian motion as a limit of a simple random walk, path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. 

Branching processes (about 6 hours)

Definition, expectation and variance of the population size, geometric branching, probability of extinction of the population.   

 

Markov chains  (about 18 hours)

Definition, homogeneous Markov chains, transition matrix, examples of Markov chains, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states, short introduction on Monte Carlo method, MCMC and search engine algorithms. 

Poisson processes (about 6 hours) 

Definition and main properties. 

Stationary processes (about 6 hours)

Definition, variance and covariance function, linear predictions, spectral theorem for autocorrelation functions, ergodic theorem.    

Recommended books:

- G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes. 3rd edn, OUP, 2001

- P. Mörters and Y. Peres. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics, 2010 

Helpful books:

- D. Williams. Probability with Martingales. CUP, 1991  

The examination consists of three questions on three different topics studied during the course. 

The students are required to answer the questions showing that they have acquired the intuition and the technical language, and that they are able to present the topics and the proofs with all necessary details.