# Mathematics and statistics (2014/2015) Course code
4S02690
Credits
12
Coordinator
Simone Ugolini
MAT/05 - MATHEMATICAL ANALYSIS
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Matematica 8 I sem. Simone Ugolini
Statistica 4 I sem. Federico Di Palma

### Lesson timetable

I sem.
Activity Day Time Type Place Note
Matematica Tuesday 3:30 PM - 6:30 PM lesson Lecture Hall A from Oct 7, 2014  to Jan 30, 2015
Matematica Friday 8:30 AM - 11:30 AM lesson Lecture Hall A from Oct 3, 2014  to Jan 23, 2015
Statistica Wednesday 1:30 PM - 4:30 PM lesson Lecture Hall F from Oct 15, 2014  to Jan 30, 2015

### Learning outcomes

Module: MATHEMATICS.
-------
This course aims at providing the students with the mathematical tools (set-theoretic and algebraic
structures, differential and integral calculus in one or several real variables, ordinary differential
equations) whose knowledge is indispensable for the achievement of the degree. A particular
attention is paid to the concrete application of the learned notions.

Module: STATISTICS.
-------
This course aims to provide the students with the fundamental of descriptive statistics, inferential statistics and probability theory.

### Syllabus

Module: MATHEMATICS
-------
1) Some notions of set theory.
2) The complete ordered field of the real numbers.
3) Euclidean distance and induced topology on the real line. Absolute value of a real number.
4) Cartesian plane.
5) Real functions of one real variable.
6) Polynomials and polynomial functions. Power, exponential and logarithmic functions. Trigonometric functions.
7) Limit of a function of one real variable.
8) Continuity of a function of one real variable at one point. Fundamental theorems on continuos functions.
9) Derivative of a function. Derivation rules. Fundamental theorems on differentiable functions.
10) Monotonicity of a function. Local and global minima and maxima of a function.
11) Convex functions.
12) Riemann integral. Integration rules. Improper integrals.
13) Ordinary differential equations. The separable and the linear case.
14) Linear algebra. Matrices and operations on them. Determinant of a square matrix.
15) Distance between two points in the plane and geometrical loci. Conics.
16) Functions of more variables. Level curves and level sets.
17) Topology in R^2. Continuity of a function of 2 variables.
18) Differentiable functions of 2 variables. Partial derivatives.
19) Local and global minima and maxima of a function of more variables.

Module: STATISTICS
-------
Part I) descriptive statistics.
Univariate statistics: main chart (pie chart, bar chart, histogram e box-plot), measures of location (mean, mode and median), measure of spread (range, interquartile range, variance, standard deviation), measure of asymmetry (third moment, skewness index, Pearson's skewness coefficient) measure of kurtosis (fourth moment, kurtosis, excess kurtosis).
Bivariate statistics: main representations (contingency tables e shattered plots), main measures (mean, variance and covariance), correlation analysis (linear regression and Pearson's correlation coefficient).

Part II) Probability theory
Probability: probability definition (classic and modern), event taxonomy (independent events, mutually exclusive events, complementary event, union event and intersection event). Conditional probability. Probability of notable events.
Random variables: discrete random variable (discrete probability distribution, expected value and variance), continuous random variable (probability density function, expected value and variance), main continuous distributions (uniform, gaussian, standard normal and chi-square).main discrete distributions (binomial and Bernoulli), central limit theorem, Chebyshev’s inequality, convergence in law of random variables and limit random variable.

Part III) Inferential Statistics.
Estimation theory: estimation problem, main properties of an estimator (unbiased, consistency and efficiency). point estimation (expected value and variance), interval estimation (expected value and variance).
Hypothesis test: Problem statement (first type and second type error, theoretical distribution), testing process, chi-square based independence test.

### Assessment methods and criteria

Module: MATHEMATICS
-------
Written exam.

Module: STATISTICS
-------
Written exam.

 Activity Author Title Publisher Year ISBN Note Matematica Guerraggio, A. Matematica per le scienze con MyMathlab (Edizione 2) Pearson 2014 9788871929415
 Title Format (Language, Size, Publication date) 0.1 - informazioni sul corso pdf (it, 94 KB, 02/11/14) 0.2 - Introduzione al corso e dizionario minimale pdf (it, 25 KB, 02/11/14) 1.1 - Statistica descrittiva I - Serie monovariate: principali rappresentazioni e definizioni di frequenze (versione stampabile) pdf (it, 125 KB, 02/11/14) 1.2 - Statistica descrittiva II - Serie monovariate: principali indici sintetici (posizione, variabilità, simmetria e curtosi), outlier e box-plot. pdf (it, 605 KB, 02/11/14) 1.3 - Statistica descrittiva III - Serie bivariate: principali rappresentazioni tabellari e grafche, indici sintetici e regressione. pdf (it, 306 KB, 26/11/14) 2.1 - Probabilità I - Definzioni introduttive, calcolo delle probabilità: definzioni (frequentistica, classica ed assiomatica), probabilitàcondizionata pdf (it, 202 KB, 02/11/14) 2.2 - Probabilità II - Variabili Casuali Discrete: definizioni introduttive, valore atteso, varianza, principali vv.cc. pdf (it, 322 KB, 19/11/14) 2.3 - Probabilità III - Variabili Casuali Continue: principali indici sintetici, principali vv.cc., teorema del limite centrale, convergenza in legge. pdf (it, 513 KB, 19/11/14) 3.1 - Inferenza I - Teoria della stima: definizioni di base, proprietà di uno stimatore, stima puntuale e per intervallo di valore atteso e varianza pdf (it, 252 KB, 19/11/14) 3.2 - Inferenza II - Test di ipotesi: principi generali, test sul valore atteso, test di aderenza alla distribuzione, test di indipendenza di Pearson, p-value pdf (it, 168 KB, 03/02/15) 4.1 - Errata corrige del 19 Nov pdf (it, 52 KB, 19/11/14) 4.2 - Errata corrige del 26 Nov pdf (it, 48 KB, 26/11/14) 4.3 - Errata corrige del 12 Dic pdf (it, 57 KB, 12/12/14) 6.2 - Raccolta Temi d'esame pdf (it, 755 KB, 15/12/14) 6.3 - Appello del 4-Febbraio Risolto pdf (it, 104 KB, 17/06/15) 6.4 - Appello del 18-Febbraio Risolto pdf (it, 124 KB, 19/02/15) 6.5 - Appello del 8-Luglio Risolto pdf (it, 102 KB, 14/07/15) 6.6 - Appello del 9-Settembre Risolto pdf (it, 131 KB, 10/09/15)