
Course unit
ANALYTICAL AND STOCHASTIC MATHEMATICAL METHODS FOR ENGINEERING
INP5070357, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
6.0 
Core courses 
MAT/06 
Probability and Mathematical Statistics 
6.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
12.0 
96 
204.0 
No turn 
Prerequisites:

Differential and Integral Calculus, Basic Linear Algebra. 
Target skills and knowledge:

The goal of the course is to introduce analytical and probabilistic tools and methods of paramount relevance in applied engineering modeling.
The student is expected to solve complex problems involving advanced analytical and probabilistic mathematical tools, including the ability to determine the appropriate setting for applied problems.
The course is preparatory for all successive courses. 
Examination methods:

Exam is written, it consists in a set of applied problems and simple abstract or theoretical questions. 
Assessment criteria:

 Proper application of tools and methods presented in the course
 Precision and care in arguing
 Ability in abstract arguments 
Course unit contents:

Analytical Methods
1. Lebesgue Measure and Integral
2. Elements of Abstract Measure and Integral
3. Normed Spaces and Banach spaces
4. Inner product spaces, Hilbert spaces
5. Elements of Harmonic Analysis: Fourier series and integrals.
Probabilistic Methods
6. Introduction to probability — probability spaces, axioms of probability, conditional probabilities, independence of events.
7. Random variables (discrete and continuous) — definition, expectation and moments. Examples of random variables and applications, with a focus on Gaussian random variables.
8. Random vectors.
9. Characteristic function.
10. Convergence of random variables: weak, in probability, in L^p, almost sure.
11. The law of large numbers and the central limit theorem with applications.
12. Conditional expectation.
13. Martingales in discrete time. 
Planned learning activities and teaching methods:

Lectures supported by exercises. 
Additional notes about suggested reading:

All teacher lecture notes are made available to the students through the moodle platform. Further material such as journal articles or lecture notes are also made available in the same fashion. Finally, indications are provided on URL sites where supplementary interesting material is available.
Sheldon Ross, A first course in Probability, Pearson, 1976. Ninth edition
Erhan Cinlar, Probability and Stochastics, Springer, 2011. 
Textbooks (and optional supplementary readings) 

Innovative teaching methods: Teaching and learning strategies
 Lecturing
 Problem based learning
 Interactive lecturing
 Questioning
 Problem solving
 Loading of files and pages (web pages, Moodle, ...)
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)
 One Note (digital ink)
 Camtasia (video editing)
 Latex
 Mathematica
 Matlab

