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

• Probability spaces and their elementary properties

• Various definitions of probability (just a hint)

• Random variables and their distributions

• Independence

• Probability measures on R and R^n (including distribution functions, and, in case of R, discrete, singular continuous, and absolutely continuous probability laws)

• List of the main probability measures on R and normal distribution on R^n

• Transform of random variables

• Moments

• Characteristic functions

• Conditional distributions

• Convergence of random variables

• Laws of large numbers

• Central limit theorems

• Infinitely divisible and stable laws

Testi/Bibliografia

In order to prepare the exam, the notes are (more than) enough, provided obviously they are taken in a exhaustive and correct way. For those who don't like notes, and/or for those who want to deepen some topics, the following text books are suggested:

Bertsekas D.P. and Tsitsikli J.N., Introduction to Probability, 2nd Edition, ISBN: 978-1-886529-23-6

Grimmett G. and Stirzaker D. (2001) Probability and random processes, Oxford University Press.

Dall'Aglio G. (1987) Calcolo delle probabilita', Zanichelli.

Metodi didattici

Regular lectures and tutorials

Modalità di verifica e valutazione dell'apprendimento

One-hour-a-half written exam, consisting of 3/4 exercises, followed by an oral examination. The exercises are obvious versions of the ones discussed in class, and attain to elements of the syllabus covered by the course lectures. The aim of the written test is to assess the student's ability to use definitions, properties and theorems of probability theory when facing with some simple problems. Finally, the oral part of the exam is subjected to the overcoming of the written part.

Strumenti a supporto della didattica

Notes and the text-books quoted above

Orario di ricevimento

Consulta il sito web di Pietro Rigo