28171 - Probability II

Learning outcomes

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

Course contents

• 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

Readings/Bibliography

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

Teaching methods

Regular lectures and tutorials

Assessment methods

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/)

Teaching tools

Lecture notes and exercises with solutions

Office hours

See the website of Alberto Lanconelli