Statistical Theory and Mathematics
A.Y. 2026/2027
Learning objectives
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.
Expected learning outcomes
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.
Lesson period: First four month period
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First four month period
More detailed directions on the teaching modalities for the academic year 2026/27 will be given in the following months, based on the evolution of the sanitary situation.
Prerequisites for admission
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.
Assessment methods and Criteria
The exam will consist in two written tests, one for the modulus Statistical Theory and one for the modulus Mathematics.
The tests will be based of open-ended and multiple answers questions, including 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.
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.
It is mandatory to pass the exams of both modules by the end of the academic year. After this time, both modules must be taken and passed again.
The tests will be based of open-ended and multiple answers questions, including 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.
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.
It is mandatory to pass the exams of both modules by the end of the academic year. After this time, both modules must be taken and passed again.
Statistical Theory
Course syllabus
1. Background
a. Probability spaces, r.v.'s, p.d.f. and c.d.f., expectation, variance, covariance.
b. Law of large numbers and central limit theorems
2. Point estimation
a. Method of moments
b. Maximum likelihood estimators
3. Properties of the estimators
a. Unbiasedness, efficiency, asymptotic normality
b. UMVU estimators
c. Cramer-Rao inequality
d. 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
e. Asymptotic tests based of likelihood function
a. Probability spaces, r.v.'s, p.d.f. and c.d.f., expectation, variance, covariance.
b. Law of large numbers and central limit theorems
2. Point estimation
a. Method of moments
b. Maximum likelihood estimators
3. Properties of the estimators
a. Unbiasedness, efficiency, asymptotic normality
b. UMVU estimators
c. Cramer-Rao inequality
d. 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
e. Asymptotic tests based of likelihood function
Teaching methods
Lecture-based teaching and exercise sessions. Homework sessions to be completed
Teaching Resources
Main reference:
G. Casella and R.L. Berger, Statistical inference, second edition, Cengage ed.
Additional references:
- R.W.Keener, Theoretical Statistics. Topics for a core course. Springer, 2010
- A. Azzalini. Statistical Inference Based on the likelihood. Taylor & Francis
- G.G.Roussas, A course in mathematical statistics, Academic Press, 1997
- Trosset M.W., An introduction to statistical inference and its applications with R, CRC Press, 2009.
G. Casella and R.L. Berger, Statistical inference, second edition, Cengage ed.
Additional references:
- R.W.Keener, Theoretical Statistics. Topics for a core course. Springer, 2010
- A. Azzalini. Statistical Inference Based on the likelihood. Taylor & Francis
- G.G.Roussas, A course in mathematical statistics, Academic Press, 1997
- Trosset M.W., An introduction to statistical inference and its applications with R, CRC Press, 2009.
Mathematics
Course syllabus
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).
Case Study: SVD decomposition
Graph Theory. Basics: nodes, edges, directed/undirected, weighted/unweighted. Representations: adjacency list, adjacency matrix, sparse representation, Laplacian (brief). Core graph concepts and simple algorithms : degree, paths, distance, connected components, diameter (concepts), BFS/DFS (intuition and complexity).
Case Study: Page Rank
Calculus. Real functions on Rn continuity, differentiability, gradient, Hessian matrix, linearization (tangent plane). Optimization. First order conditions for unconstrained problems. Constrained optimization: equality constraints and Lagrange Multipliers. Inequality constraints. Linear programming.
Case Study: minimization of a loss function
Discrete and continuous dynamical systems with applications.
Case Study: simple ecological model
Case Study: SVD decomposition
Graph Theory. Basics: nodes, edges, directed/undirected, weighted/unweighted. Representations: adjacency list, adjacency matrix, sparse representation, Laplacian (brief). Core graph concepts and simple algorithms : degree, paths, distance, connected components, diameter (concepts), BFS/DFS (intuition and complexity).
Case Study: Page Rank
Calculus. Real functions on Rn continuity, differentiability, gradient, Hessian matrix, linearization (tangent plane). Optimization. First order conditions for unconstrained problems. Constrained optimization: equality constraints and Lagrange Multipliers. Inequality constraints. Linear programming.
Case Study: minimization of a loss function
Discrete and continuous dynamical systems with applications.
Case Study: simple ecological model
Teaching methods
Teaching methods frontal lectures and practicals (some exercise will involve the presentation of case studies using software tools).
Homework sessions to be completed online are planned
Homework sessions to be completed online are planned
Teaching Resources
(not mandatory)
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, 20163.
3. E. Salinelli, F. Tomarelli, Discrete-Dynamical Models, Springer, 2014, ISBN: 978-3-319-04.
Notes/slides of the teachers
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, 20163.
3. E. Salinelli, F. Tomarelli, Discrete-Dynamical Models, Springer, 2014, ISBN: 978-3-319-04.
Notes/slides of the teachers
Modules or teaching units
Mathematics
MATH-05/A - Numerical Analysis - University credits: 6
Lessons: 40 hours
Professor:
Naldi Giovanni
Statistical Theory
STAT-01/A - Statistics - University credits: 6
Lessons: 40 hours
Professor:
Leorato Samantha
Professor(s)
Reception:
Monday 14:30-16 or Thursday 9-10:30 (online). Email is required
Room 32 third floor