Lecturer: Benedetta Ferrario  

Course name: Mathematical analysis (complements) and statistics
Course code: 500786
Degree course: Ingegneria Industriale
Disciplinary field of science: MAT/05, MAT/06
University credits: CFU 9
Course website: http://www-dimat.unipv.it/ferrario

Specific course objectives

This is a second course in calculus and a first course in mathematical probability with an introduction to statistical inference. It includes series, vector analysis, multiple integrals, line and surface integrals, the integral theorems of vector calculus; moreover, the calculus of probability, combinatorial analysis, independence, conditional probability, Bayes’ theorem, random variables, expectation, variance, distribution functions, law of large numbers and central limit theorem; interval estimation.

Course programme

Mathematical Analysis

• Series; absolute and simple convergence; series with positive terms; special series. Convergence results. Power series; derivation and integration. Taylor expansion.
• Calculus for functions of several variables. Limits, continuity, partial derivatives, gradient, differentiability, Hessian; stationary points and their classification. Taylor's formula. Calculus for vector functions; Jacobian.
• Multiple integrals. Two dimensional integrals; change of coordinates, polar coordinates, techniques of integration. Three dimensional integrals: spherical or cylindrical coordinates; evaluating the integral by the slice method or the line method.
• Line and surface integrals. Parametric equations of a line; tangent line; arc lenght. Parametric equations of a surface; tangent plane; surface area; surface of revolution. Line integrals of scalar fields and of vector fields. Conservative vector fields. The differential operators curl and div. Surface integrals. Green's theorem; Stokes' theorem; divergence theorem.

Statistics

• Definition of probability. Conditional probability; Bayes' theorem. Independence. Mathematical expectation, variance. Random variables; discrete and continuous. Chebyshev inequality. Law of large numbers. Central limit theorem. Student's t-distribution and chi-square distribution.
• Inferential statistics; confidence intervals for the mean value and the variance. Linear regression.

Course entry requirements

Analisi Matematica I, Geometria e Algebra.

Course structure and teaching

Lectures (hours/year in lecture theatre): 50
Practical class (hours/year in lecture theatre): 35
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

C. Canuto e A. Tabacco . Analisi Matematica II . Springer, 2008.

P. Baldi. Introduzione alla probabilita` con elementi di statistica. McGraw-Hill.

Testing and exams

The exam consists of two parts: written and oral. Admission to the oral exam only if the result of written exam is not less than 16/30. Both exams must be completed in the same session. The written test takes places on the day of the beginning of the exam session; the oral exam will start few days later.