Probability and Mathematical Statistics 1

A.Y. 2018/2019
Overall hours
Learning objectives
1) The course provides basic notions and methods of probability theory, based on measure theory. Fundamentals will be taught to students at the level required for their further study in mathematics.
2) The course aims at giving precise and systematic knowledge on real and vector random variables (laws, moments, independence, conditional laws etc.). This is required for an appropriate understanding of probabilistic models and for further study in statistics, stochastic processes and stochastic calculus.
3) Within the course the most important limit theorems of the theory of probability will be carefully presented (laws of large numbers and central limit theorem). Basic applications will be given, in particular to statistics.
4) In the laboratory, fundamental notions will be given on generation of random numbers with preassigned distributions and the most basic applications of these techniques will be presented.
Expected learning outcomes
Students will acquire the basics of probability theory required to further pursue their study towards mathematics or its applications, either in mathematical statistics or in stochastic processes or in stochastic calculus and its applications.
Course syllabus and organization

Single session

Lesson period
Second semester
Course syllabus
1) Theory of probability.
Axiomatic definition of probability. Conditional probability and independence. Probability and random variables in sample space with countable outcomes. Probability on the real numbers.

2) Random variables.
Random variables and their laws. Integration with respect to a probability measure. Expectation and moments. Random vectors and multivariate laws. Independence of random variables. Examples of random variables used in basic probability models. Characteristic functions. Multivariate Gaussian laws and Gaussian samples.

3) Convergence of random variables.
Almost sure convergence, convergence in probability, convergence in mean of order p. Laws of large numbers. Weak convergence of probability measures. Characteristic functions and convergence in law. Central limit theorem. Applications to statistics.

4) Conditional expectation.
General definition of conditional expectation given a sigma-algebra. Conditional probabilities and conditional laws. Conditional laws for Gaussian vectors.

5) Laboratory.
Generation of random numbers with given laws. The method of inverse distribution function and the rejection method. Applications.

Propedeuticità consigliate

Geometria 1, Analisi Matematica 1, Analisi Matematica 2, Analisi Matematica 3.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 44 hours
Laboratories: 12 hours
Lessons: 36 hours
Turno A
Professor: Morale Daniela
Turno B
Professor: Maurelli Mario
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.
Monday 14-16 by appointment by email; other days by appointment by email
online meeting
upon appointment via email
Via Saldini 50, second floor