Probability and Statistics 2

Credits

Prerequisites

Probability and Statistics, Mathematical Analysis 2.

Scientific-disciplinary sector (SSD)

MAT/06 Probability and Mathematical Statistics.

Examination method

Oral exam including the solution of an exercise.

Learning
objectives

The course aims to present the fundamental elements of measure theory in the probabilistic setting, examining some specific topics in depth, and to provide the theoretical principles underlying certain methods of inferential statistics and the conditions under which they apply.

Syllabus

Completion of a probability space. Random variables. Representation theorem. The concept of stochastic independence and 0-1 laws. Integration of measurable functions and moments. Notable inequalities and their interpretation in terms of moments. The characteristic function of a random variable. Types of convergence of a sequence of random variables and asymptotic theorems (strong law of large numbers). The n-dimensional extension. Conditional expectation with respect to a sigma-algebra. Probability distributions and parametric models of particular interest in Mathematical Statistics. Point estimation (order statistics, unbiased estimators, minimum-variance estimators, asymptotic properties of estimators, efficient statistics, minimal sufficient statistics, ancillary statistics, complete statistics, methods for constructing estimators, Bayes estimators). Interval estimation.

Expected learning
outcomes

On completion of the course, students must demonstrate that they

know and understand the statements and contents of the course, have a command of the related proof techniques, and be aware of the probabilistic structure underlying statistical methods;
can apply the knowledge acquired by rigorously modelling a random phenomenon and solving it through the most appropriate methods;
can communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
can identify the most appropriate methods to analyse and solve a problem relating to the course topics and interpret the results correctly.

Learning outcomes
to be assessed

Independence in choosing appropriate techniques for solving exercises; attainment of a sufficient command of the relevant language and of the techniques used in proofs. Clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Probability and Statistics 2