#
Fundamental of mathematics and statistics

A.Y. 2015/2016

Learning objectives

This course is split into two units. The first one is essentially a course in calculus. Therefore, its aim is the introduction of the elements necessary to study real functions, differential and integral calculus with one variable. Moreover some basic topics in linear algebra are given, with the purpose of solving linear systems. The second module, an introduction to Statistics, is devoted to present some of the essential notions of the statistical analysis, useful to Natural Sciences, with a special attention to those aspects that are of interest for a better understanding of the analysis of experimental data.

Expected learning outcomes

Capacity of analyzing simple graphics of functions and diagrams of statistic data and of elaborating information from this. Capacity of constructing simple graphics of functions and diagrams of statistic data. Ability in solving problems involving linear systems. Ability in associating with simple random situations the correct probabilistic/statistic model and in applying it to data. Capability of correctly applying basic statistic tools and tests.

**Lesson period:** year
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

year

**Unita' didattica: matematica**

**Course syllabus**

Real numbers. Linear algebra: vectors, matrices, systems of linear equations.

Differential and integral calculus with one variable. Functions and their graphs. Limits. Derivates. Integrals. Rules of derivation and integration. Maxima and minima of functions. Convex functions. Taylor formula. Integrals.

Differential and integral calculus with one variable. Functions and their graphs. Limits. Derivates. Integrals. Rules of derivation and integration. Maxima and minima of functions. Convex functions. Taylor formula. Integrals.

**Unita' didattica: statistica**

**Course syllabus**

Descriptive statistics. Types of Data. Frequency tables. Pictures of Data. Position indexes: mean, median, mode, percentiles, quartiles. Variation indexes: variance, standard deviation. Probability. Axioms of a probability measure. Classical probability. Combinatorial calculus: counting the possible configurations of n elements in k places. Conditional probability. Total probability theorem. Bayes formula. Finite and continuous random variables (r.v.). R.v. law. Probability function of a finite r.v..

Density and distribution functions of a continuous r.v..

Binomial distributions. Continuous distributions: normal distribution. Mean, variance, standard deviation of a finite r.v. and of a continuous r.v. and their properties. Central limit theorem. Sums of independent random variables and normal approximation. Confidence intervals for the mean of a normal distribution, with standard deviation known and with standard deviation unknown. Binomial proportion confidence interval.Statistical inference: populations and samples, the basic idea of statistical test. Power, protection and the detection of differences: Type I and Type II error. Comparisons for enumeration data: Fisher's exact test, the χ2 test, contingency tables. Comparisons of two sample means: Student's t test. Comparisons of any number of sample means: the Analysis of Variance (ANOVA). One-way ANOVA: the completely random design, the randomised complete block design. Regression analysis: the linear regression model and equation, tests of hypotheses. ANOVA of regression.

Density and distribution functions of a continuous r.v..

Binomial distributions. Continuous distributions: normal distribution. Mean, variance, standard deviation of a finite r.v. and of a continuous r.v. and their properties. Central limit theorem. Sums of independent random variables and normal approximation. Confidence intervals for the mean of a normal distribution, with standard deviation known and with standard deviation unknown. Binomial proportion confidence interval.Statistical inference: populations and samples, the basic idea of statistical test. Power, protection and the detection of differences: Type I and Type II error. Comparisons for enumeration data: Fisher's exact test, the χ2 test, contingency tables. Comparisons of two sample means: Student's t test. Comparisons of any number of sample means: the Analysis of Variance (ANOVA). One-way ANOVA: the completely random design, the randomised complete block design. Regression analysis: the linear regression model and equation, tests of hypotheses. ANOVA of regression.

Unita' didattica: matematica

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

Practicals: 32 hours

Lessons: 32 hours

Lessons: 32 hours

Professors:
Mazza Carlo, Rizzo Ottavio Giulio

Unita' didattica: statistica

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

Practicals: 32 hours

Lessons: 16 hours

Lessons: 16 hours

Professors:
Biganzoli Elia, Pacifici Emanuele

Professor(s)

Reception:

Wednesday, 13.30-15.30 (for scheduling purposes, interested people are invited to write an email)

Dipartimento di Matematica, II floor, Room 2093