28171 - PROBABILITY II

Conoscenze e abilità da conseguire

By the end of the course the student should know the basic theory of multidimensional random variables and sequences of random variables. In particular the student should be able: - to derive the distribution of transformed random variables - to derive joint, conditional and marginal probability density functions - to state the definition and recall the properties of multivariate normal distributions – to investigate converge properties of sequences of random variables

Contenuti

• Useful inequalities and Lebesgue spaces
  
• Convergence concepts for sequences of random variables
  
• Borel-Cantelli lemma and applications
• Characteristic function of a random variable
  
• Weak and strong law of large numbers
  
• Central limit theorem
• Random vectors and multidimensional Gaussian distribution

Testi/Bibliografia

Lecture notes. Suggested readings:

• Kai Lai Chung, A course in Probability Theory - III edition, Aca-
  demic Press, San Diego, 2001
• Rick Durrett, Probability: Theory and Examples - IV edition, Cam-
  bridge University Press, Cambridge, 2013

Metodi didattici

Regular lectures and tutorials

Modalità di verifica e valutazione dell'apprendimento

One-hour written exam, articulated in a series of 3 exercises each with a maximum grade of 10 points, followed by an oral examination. Every exercise attains to elements of the syllabus covered during the course lectures. Online exams will be supported by the softwares Teams, Zoom and EOL (https://eol.unibo.it/)

Strumenti a supporto della didattica

Lecture notes and exercises with solutions

Orario di ricevimento

Consulta il sito web di Alberto Lanconelli