Statistical Theory and Mathematics
A.Y. 2023/2024
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 trimester
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 trimester
More detailed directions on the teaching modalities for the academic year 2023/24 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 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.
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, the tests of both modules must be taken and passed again.
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.
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, the tests of both modules must be taken and passed again.
Statistical Theory and Mathematics-Module Statistical Theory
Course syllabus
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
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
Teaching methods
Frontal lectures and exercises.
Teaching Resources
1. G. Casella and R.L. Berger, Statistical inference, second edition, Cengage ed.
2. R.W.Keener, Theoretical Statistics. Topics for a core course. Springer, 2010
3. G.G.Roussas, A course in mathematical statistics, Academic Press, 1997
4. Trosset M.W., An introduction to statistical inference and its applications with R, CRC Press, 2009.
2. R.W.Keener, Theoretical Statistics. Topics for a core course. Springer, 2010
3. G.G.Roussas, A course in mathematical statistics, Academic Press, 1997
4. Trosset M.W., An introduction to statistical inference and its applications with R, CRC Press, 2009.
Statistical Theory and Mathematics-Module 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).
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.
Teaching methods
Frontal lectures
Teaching Resources
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
Statistical Theory and Mathematics-Module Mathematics
MAT/08 - NUMERICAL ANALYSIS - University credits: 6
Lessons: 40 hours
Professors:
Benfenati Alessandro, Liuzzi Danilo
Statistical Theory and Mathematics-Module Statistical Theory
SECS-S/01 - STATISTICS - University credits: 6
Lessons: 40 hours
Professor:
Leorato Samantha
Educational website(s)
Professor(s)
Reception:
Next office hours: Thursday May the 2nd 10:30-13:30.
Room 32 third floor
Reception:
Wednesday 10:30-13:30 am
Room 32, Via Conservatorio 7, DEMM, 3rd floor