Probability and Statistics

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Probability and Statistics

Credits

9

Prerequisites

Mathematical Analysis 2.

Scientific-disciplinary sector (SSD)

MAT/06 Probability and Mathematical Statistics.

Examination method

Oral exam.

Learning
objectives

The course aims to provide students with a mathematically rigorous presentation of the basic contents of these disciplines, through a precise definition of the concepts and a careful study of the results and their proofs.

Syllabus

The empirical law of chance. Empirical frequency and probability. A priori probability. Geometric probability (outline). Definition of subjective probability. Elements of combinatorics. The measurable space and the structure of events. Bernoulli sample space. Sequences and their limits. The probability measure. Independence of events. Borel-Cantelli lemma. 0-1 law. Conditional probabilities. Sets of alternatives. Law of alternatives. Bayes’ theorem. Definition of random variable. The distribution function and its properties. Notable discrete random variables: Bernoulli, binomial, geometric, uniform, degenerate, Poisson (as a limit of binomials). Notable absolutely continuous random variables: uniform, exponential, normal. Transformations of random variables. Multidimensional random variables. Independence of random variables. Sums, products and ratios of random variables. Moments of one-dimensional random variables and their properties. Moments of functions of random variables. Moments of random vectors and the case of independent random variables. Two-dimensional vectors. Properties of the mean and variance. Covariance and correlation coefficient. Standardised random variables. Probability generating function and moment generating function and their properties. Convergence in law or in distribution. Convergence in probability. Čebyšëv’s inequality. Bernoulli’s theorem. Schwarz’s inequality and Markov’s inequality. Convergence theorems. Sampling and special distributions. Point estimation and properties of estimators. Methods for constructing estimators: method of moments and maximum likelihood method, with examples.

Expected learning
outcomes

At the end of the course, students must demonstrate that they

  • know and understand the basic topics and the main tools of probability theory and statistics, and are able to identify and understand a probabilistic model;
  • are able to apply the knowledge acquired by rigorously schematising a random phenomenon and solving it by identifying the most appropriate methods of probability and statistics;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Assessment of knowledge of the course contents, mastery of the related language and of the techniques used in the proofs, and rigour of presentation. The ability to use the notions acquired to tackle theoretical questions of considerable complexity is a further assessment criterion.