28171 - Probability II

Learning outcomes

By the end of the course the student should know the basic theory of multidimensional random variables and independent 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 explain the role played by Wishart, Lamba and T2 distributions in multivariate normal inference

Course contents

Continuos and discrete random variables. The moment-generating function. Functions of a random variable. Expected value and variance of a function of a random variable. Bivariate distributions. The bivariate normal distribution and regression.  Random vectors. The joint moment-generating function. Sequences of random variables: sums and averages of random variables. Convergence of random variables. The central limit theorem. Functions of a random vector. The multivariate normal distribution.

Readings/Bibliography

Multivariate probability, John H. McColl, 2004, Arnold publisher, London 
Probability, an Introduction, G. Grimmett & D. Welsh, 2014, Oxford University Press (chapters 6,7,8)

Teaching methods

Lectures and tutorials

Assessment methods

The learning assessment is composed by a written test lasting 2 hours, followed by an oral examination. The written test is aimed at assessing the student's ability to use the definitions, properties and theorems of probability theory in solving theoretical exercises. During the written exam, students can make use of a formulary and a calculator. The written test consists of 3 or 4 exercises articulated in several points with a final grade out of thirty.

Office hours

See the website of Cinzia Viroli