Statistical theory and mathematics
A.A. 2022/2023
Obiettivi formativi
The purpose of the course is that students learn the main mathematical, statistical and computational tools needed to approach a data science problem. The course serves mostly to refresh students' knowledge, and to ensure that all students have a common mathematical and statistical background.
Risultati apprendimento attesi
At the end of the course, students will be able to formalize real world problems in a mathematical way and to implement the appropriate statistical inference methods.
Periodo: Primo trimestre
Modalità di valutazione: Esame
Giudizio di valutazione: voto verbalizzato in trentesimi
Corso singolo
Questo insegnamento non può essere seguito come corso singolo. Puoi trovare gli insegnamenti disponibili consultando il catalogo corsi singoli.
Programma e organizzazione didattica
Edizione unica
Responsabile
Periodo
Primo trimestre
Prerequisiti
The students are requested to have attended
a) a basic Calculus course and a basic Linear Algebra course.
b) an introductory course in probability and statistics, including the concepts of probability space, random variable, distribution and basic descriptive statistics.
a) a basic Calculus course and a basic Linear Algebra course.
b) an introductory course in probability and statistics, including the concepts of probability space, random variable, distribution and basic descriptive statistics.
Modalità di verifica dell’apprendimento e criteri di valutazione
For the students of Data Science for Economics (DSE) the exam will consist in two written tests, one for the modulus Statistical Theory and one for the modulus Mathematics.
The students of Environmental Change and Global Sustainability (ECGS) must attend only the modulus Mathematics and thus take only the exam of the modulus Mathematics.
The tests will include open-ended and multiple answers questions, both theoretical and short exercises, with the aim of assessing the student's ability to solve simple problems in applied mathematics and/or in statistics.
For the students of DSE: the global exam is passed if the tests of the first and second module are passed, that is if in each test the student deserved at least 18/30.
Final marks are given using the numerical range 0-30 and are composed as the mean of the grades of the two modules. The final grade can be increased of up to 3 points if the student additionally develops and presents to the teachers a project in applied math or in data analysis. The project is not mandatory and can be developed by a group composed by 1 to 5 students.
It is mandatory to pass the exams of both modules by the end of the academic year. After this time, the tests of both modules must be taken and passed again.
For the students of ECGS: the exam is passed if the test of the module Mathematics is passed, with at least 18/30. The final grade can be increased of up to 3 points if the student develops and presents to the teachers a project in applied math. The project is not mandatory and can be developed by a group composed by 1 to 5 students.
The students of Environmental Change and Global Sustainability (ECGS) must attend only the modulus Mathematics and thus take only the exam of the modulus Mathematics.
The tests will include open-ended and multiple answers questions, both theoretical and short exercises, with the aim of assessing the student's ability to solve simple problems in applied mathematics and/or in statistics.
For the students of DSE: the global exam is passed if the tests of the first and second module are passed, that is if in each test the student deserved at least 18/30.
Final marks are given using the numerical range 0-30 and are composed as the mean of the grades of the two modules. The final grade can be increased of up to 3 points if the student additionally develops and presents to the teachers a project in applied math or in data analysis. The project is not mandatory and can be developed by a group composed by 1 to 5 students.
It is mandatory to pass the exams of both modules by the end of the academic year. After this time, the tests of both modules must be taken and passed again.
For the students of ECGS: the exam is passed if the test of the module Mathematics is passed, with at least 18/30. The final grade can be increased of up to 3 points if the student develops and presents to the teachers a project in applied math. The project is not mandatory and can be developed by a group composed by 1 to 5 students.
Module Statistical Theory
Programma
1. Background
a. Probability spaces, r.v.'s, p.d.f. and c.d.f., expectation, variance, covariance.
b. Basic inequalities (Markov, Chebyshev)
c. Law of large numbers
d. Moments generating function
e. Central Limit Theorem
2. Point estimation
a. Method of moments
b. Maximum likelihood estimators
3. Properties of the estimators
a. Unbiasedness, efficiency, asymptotic normality
b. Sufficiency and Fisher-Neyman factorisation
c. Completeness
d. UMVU estimators
e. Cramer-Rao inequality
f. efficient estimators and Fisher information
4. Confidence intervals
5. Hypothesis testing
a. Power of a test and UMP tests
b. Neyman-Pearson lemma
c. MLR method to find a test
d. Main classical parametric tests on one sample or two samples
6. Introduction to Bayesian estimate
7. Examples and case studies in R
a. Probability spaces, r.v.'s, p.d.f. and c.d.f., expectation, variance, covariance.
b. Basic inequalities (Markov, Chebyshev)
c. Law of large numbers
d. Moments generating function
e. Central Limit Theorem
2. Point estimation
a. Method of moments
b. Maximum likelihood estimators
3. Properties of the estimators
a. Unbiasedness, efficiency, asymptotic normality
b. Sufficiency and Fisher-Neyman factorisation
c. Completeness
d. UMVU estimators
e. Cramer-Rao inequality
f. efficient estimators and Fisher information
4. Confidence intervals
5. Hypothesis testing
a. Power of a test and UMP tests
b. Neyman-Pearson lemma
c. MLR method to find a test
d. Main classical parametric tests on one sample or two samples
6. Introduction to Bayesian estimate
7. Examples and case studies in R
Metodi didattici
Frontal lectures and exercises.
The exercises classes will be offered separately for the students of DSE and for the students of ECGS.
The exercises classes will be offered separately for the students of DSE and for the students of ECGS.
Materiale di riferimento
1. R.W.Keener, Theoretical Statistics. Topics for a core course. Springer, 2010
2. G.G.Roussas, A course in mathematical statistics, Academic Press, 1997
3. Trosset M.W., An introduction to statistical inference and its applications with R, CRC Press, 2009.
4. Notes of the teacher
2. G.G.Roussas, A course in mathematical statistics, Academic Press, 1997
3. Trosset M.W., An introduction to statistical inference and its applications with R, CRC Press, 2009.
4. Notes of the teacher
Module Mathematics
Programma
Linear Algebra and applications. Real vector spaces. Linear combination, linear dependence and independence. Basis and dimension in Rn. Algebra of vectors, inner product and norm. Matrix algebra (inverse, rank, derivatives, eigenvalues, singular value decomposition).
Calculus. Real functions on Rn (continuity, differentiability, implicit function theorem, basic fixed point theorem, gradient).
Optimization. First and Second order conditions for unconstrained problems. Constrained optimization: equality constraints and Lagrange Multipliers. Inequality constraints. Linear programming.
Discrete and continuous dynamical systems with applications.
Examples and case studies in R.
Calculus. Real functions on Rn (continuity, differentiability, implicit function theorem, basic fixed point theorem, gradient).
Optimization. First and Second order conditions for unconstrained problems. Constrained optimization: equality constraints and Lagrange Multipliers. Inequality constraints. Linear programming.
Discrete and continuous dynamical systems with applications.
Examples and case studies in R.
Metodi didattici
Frontal lectures
Materiale di riferimento
1. David C. Lay, Steven R. Lay and Judi J. McDonald, Linear Algebra and Its Applications, Pearson, 2016
2. K. Sydsaeter, P. Hammond, A. Strom, A. Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2016
3. E. Salinelli, F. Tomarelli, Discrete-Dynamical Models, Springer, 2014, ISBN: 978-3-319-02290-1
4. Notes of the teachers
2. K. Sydsaeter, P. Hammond, A. Strom, A. Carvajal, Essential Mathematics for Economic Analysis, Pearson, 2016
3. E. Salinelli, F. Tomarelli, Discrete-Dynamical Models, Springer, 2014, ISBN: 978-3-319-02290-1
4. Notes of the teachers
Moduli o unità didattiche
Module Mathematics
SECS-S/01 - STATISTICA - CFU: 6
Lezioni: 40 ore
Docenti:
Micheletti Alessandra, Naldi Giovanni
Module Statistical Theory
MAT/08 - ANALISI NUMERICA - CFU: 6
Lezioni: 40 ore
Docente:
Micheletti Alessandra
Siti didattici
Docente/i
Ricevimento:
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