
Course unit
ANALYTICAL AND STOCHASTIC MATHEMATICAL METHODS FOR ENGINEERING
INP5070357, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2017/18
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 
Mode of delivery (when and how)
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Organisation of didactics
Type of hours 
Credits 
Hours of teaching 
Hours of Individual study 
Shifts 
Lecture 
12.0 
96 
204.0 
No turn 
Start of activities 
02/10/2017 
End of activities 
19/01/2018 
Prerequisites:

None 
Target skills and knowledge:

Objective
Introduce the students to the basic knowledge of mathematical and probability tools for engineers.
Outcomes
A student who has met the objectives of the course will have a comprehensive knowledge of :
• Basic mathematical tools of functional analysis and measure theory
• Basic concepts of probability 
Examination methods:

Final examination based on: written examination. 
Assessment criteria:

Critical knowledge of the course topics. Ability to present the studied material. 
Course unit contents:

1. Lebesgue Measure and Integral — Lebesgue measure and integral on R^d, limit theorems, dependence by parameters, reduction formula and change of variables. Introduction to abstract measure and integral.
2. Normed spaces — Concept of norm; uniform, L^1 and L^p norms; sequences, limits, completeness and Banach spaces. Concept of scalar product and Hilbert space; the space L^2; separable spaces and concept of orthonormal base.
3. Fundamentals of Complex Analysis — Power series in C and elementary analytic functions; concept of holomorphic function, CauchyRiemann equations, analyticity of holomorphic functions; isolated singularities; residue theorem and applications.
4. Introduction to Fourier Analysis — Fourier series: Euler formulas, convergence and sum (pointwise, uniform, L^2). Fourier transform: L^1 definition, convolution and approximate units, inversion formula. Schwarz functions and L^2 Fourier transform. Applications.
5. Introduction to probability — probability spaces, axioms of probability, conditional probabilities, independence of events.
6. Random variables (discrete and continuous) — definition, expectation and moments. Examples of random variables and applications, with a focus on Gaussian random variables.
7. Random vectors.
8. Characteristic function.
9. Convergence of random variables: weak, in probability, in L^p, almost sure.
10. The law of large numbers and the central limit theorem with applications.
11. Conditional expectation.
12. Martingales in discrete time. 
Planned learning activities and teaching methods:

Lectures supported by exercises. 
Additional notes about suggested reading:

Lecture notes relative to the Analytical Methods part will be provided.
See the reference book below for the Probability part. 
Textbooks (and optional supplementary readings) 

Sheldon Ross, A first course in Probability. : Pearson, 1976. Ninth edition


