OVERVIEW

Stochastic Processes: the theory of Markov chains is presented, both at discrete and continuous time, with particular attention to Poisson processes and queuing theory. The goal is to provide the student with the ability to model real problems of stochastic evolution in terms of Markov chains (when possible). 

Teaching contributes to the achievement of Goals 4 and 5 of Sustainable Development of the UN 2030 Agenda.

Mathematical Statistics: An introduction to the classical theory of statistical models (model identification and estimation, parametric and not parametric models, exponential models), point estimation (moment method, likelihood method and invariant estimators).

Teaching contributes to the achievement of Goals 1,4, 5, 8 of Sustainable Development of the UN 2030 Agenda.

AIMS AND CONTENT

LEARNING OUTCOMES

We want to introduce Markov chains and other simple stochastic processes in order to model and solve real problems of stochastic evolution.

AIMS AND LEARNING OUTCOMES

Stocasthic Processes: The goal is to have the student learn the language of Markov chains, so that he will be able to build an accurate model starting from real problems of stochastic evolutions taking values in a finite or countable set (the set of states). 

At the end of the course the student will have to:
- know the general theory of Markov chains, both at discrete and continuous time,          

- classify the states and determine the invariant laws with respect to the evolutions of the system,                                                          

- suitably model real situations of the queueing theory in the language of Markov chains, and be able to study the efficacy of the model. 

Mathematical Statistics: 

Learning outcomes are to formalise estimation problems (parametric and non-parametric) and statistical hypothesis testing in a rigorous mathematical framework, to formulate and apply appropriate regression models to various typologies of data sets.

At the end of the course students will be able to

• recognise estimation problems (both parametric and non parametric) in applied contexts
• formulate them in a rigorous mathematical framework
• determine estimators of model parameters and evaluate their goodness
• write definitions, statements and demonstrations and produce related examples and counterexamples.

PREREQUISITES

Stocasthic Processes: Probability.

You can find more details on Aulaweb.

Mathematical Statistics: Probability, Mathematical Analysis 1 and 2, Inferial Statistics.

TEACHING METHODS

Stochastic Processes: Teaching is done the traditional way, with lectures held at the blackboard. Expect 2 theory classes per week (4 hours) and 1 of exercises (2 hours).
At the end of the course there will be a guided full-text exercise so as to give students the opportunity to understand their degree of readiness and to clarify together possible doubts. 

Attendance is not mandatory but strongly recommended.

Mathematical Statistics: Combination of traditionals lectures and exercises.

SYLLABUS/CONTENT

Stocasthic Processes:

Discrete time Markov chains. Definition. Classification of states. Transience and recurrence criteria. Probability of absorbtion in recurrence classes. Invariant laws. Limit Theorems. Convergence to equilibrium. Applications: random walks.

Contnuous time Markov chains. Hitting time. Chapman-Kolmogorov equations. Invariant laws. Jumps chain. Born and death chains. Poisson processes.

Queueing theory.

Mathematical Statistics: 

Review of essential probability including the notion of conditional probability and multivariate normal distribution.

Statistical models and statistics|: the ideas of data sample and of statistical model, identifiability and regular models, the exponential family. Statistics and their distributions. Sufficient, minimal and sufficient, ancillary, complete statistics. The lemma of Neyman-Fisher. The Basu theorem.

Point estimators and their properties: methods to find point estimators: moment methods, least square method, maximum likelihood method, invariant estimators.

RECOMMENDED READING/BIBLIOGRAPHY

Stochastic Processes:

P. Baldi, Calcolo delle Probabilità e Statistica Matematica

W. Feller, An introduction to Probability Theory and its Applications

S. Karlin, H.M. Taylor, A First Course in Stochastic Processes.

S. Karlin, H.M. Taylor, A Second Course in Stochastic Processes.

S.M. Ross, Introduction to Probability Models.

G. Grimmett, D. Stirzaker, (2001). Probability and Random Processes. 

J.R. Norris. Markov Chains.

P. Brémaud. Markov Chains: Gibbs Fields, Montecarlo Simulation, and Queues.

Notes

Mathematical Statistics:

Text books:   

G. Casella e R.L. Berger, Statistical inference, Wadsworth 62-2002-02  62-2002-09
D. A. Freedman, Statistical Models, Theory and Practice, Cambridge 62-2009-05

L. Pace e A. Salvan, Teoria della statistica, CEDAM 62-1996-01   
M. Gasparini, Modelli probabilistici e statistici, CLUT 60-2006-08   
D. Dacunha-Castelle e M. Duflo, Probabilites et Statistiques, Masson 60-1982-18/19/26 e 60-1983-22/23/24   
A.C. Davison, Statistical Models, Cambridge University Press, Cambridge, 2003 

Suggested reading:

David J. Hand, A very short introduction to Statistics, Oxford 62-2008-05
L. Wasserman. All of Statistics, Springer 
J. Protter, Probability Essentials, Springer 60-2004-09 
S.L. Lauritzen, Graphical models, Oxford University press 62-1996-14 
D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1991

Handouts.

TEACHERS AND EXAM BOARD

Exam Board

VERONICA UMANITA' (President)

DAMIANO POLETTI

EMANUELA SASSO (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Stochastic Processes: Written test + oral test. 
To participate in the written test you must register on the UNIGE site.
The written exam is only passed by scoring greater than or equal to 18 marks out of 30. The oral examination can be taken immediately after the written test or even in subsequent exam sessions during the academic year in progress. 

The written test consists of 2 exercises, one on the discrete part and the other one on the continuous part.
The duration of the test is 3 hours and access to the course notes (including exercises done in the classroom) and handouts is allowed. 

The oral test will consist of exposition of theoretical arguments, proofs and exercises. Students in the master's degree in mathematics will also be asked for a proof (of the teacher's choice) from among those not given in class.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

Mathematical Statistics: Written and oral exam. 

For Smid students, the final grade will be given by the weighted average (over the CFUs) of the grades for the two parts of the teaching.

ASSESSMENT METHODS

Stochastic Processes: The oral examination is aimed at assessing the general understanding of the course topics and it is required that the student knows how to properly expose the concepts seen in the course, to show the main results and to solve the exercises.

Mathematical Statistics: In the written exam there are three or four exercises. Past exams with solutions are available on the websites. The oral exam consists of questions on both parts of the course. The course work done during the lab sessions might be subject of the oral exam (thus bring with you at the exams that course work). 

Exam schedule