KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will know the elementary theory of
probability and statistics, in particular the theory of discrete and
continuos random variables, and some elementary results in mathematical
statistics.
APPLYING KNOWLEDGE AND UNDERSTANDING
The student will be able to solve elementary exercises and also to
produce elementary probability models in simple practical situations.
MAKING JUDGEMENTS
The student will autonomusly recognise the elementary technique of
probability theory.
COMMUNICATION SKILLS
The student will be albe to express simple but precise concepts of
probability theory and statistics.
LEARNING SKILLS
The student will be able to read a study university textbook in probability
and statistics.

Basic knowledge of infinitesimal calculus and linear algebra. Prerequisites: Analysis 2, Geometry 1.

• Finite and countable sample spaces. Events. Sure event, impossible event, incompatible events, jointly exhaustive events. Discrete uniformly probability theory. • Indipendent events. Conditional probability under an event. Bayes theorem. Alternatively Bayes’s law. • Basic combinatorics, permutations and combinations. Pascal triangle. Binomial expansion. • Discrete random variables. Distribution functions of discrete random variables. Discrete random vectors. Joint distribution and marginal distributions of a discrete random vector. Expected value and variance of a discrete random variable. • Indipendent discrete random variables. • Discrete random variables: uniform distribution, Bernoulli distribution, binomial distribution, geometric distribution, Poisson distribution, ipergeometric distribution. Connection between binomial and Poisson distribution. • Sigma-algebras. Kolmogorof axioms of probability theory. Borel sets. Random variables. Real valued random variables and real valued random vectors. Definition and main properties of the distribution function of a real valued random variable. Joint distribution function of a real valued random vector. • Continuos random variables. Properties of the density function of a continuous random variable. Continuous 2-dimensional random vectors. Joint distribution and marginal distributions of a continuous 2- dimensional random vector. Expected value and variance of a continuous random variable. • Indipendent continuous random variables. • Moment-generating function. • Continuous random variables: uniform continuous distribution, gamma distribution, exponential distribution, normal distribution, Chi-squared distribution. • Convergence of random variables. Almost sure convergence, convergence in probability, convergence in distribution. Weak law of large numbers. Central limit theorem. Approximate normality. • Observations and samples. Unknown parameters. Statistics. Sample mean, sample variance. Populations moments. • Statistical inference. Likelihood function. Parameters of a statistical model. Method of estimating parameters. Point estimators. Maximum likelihood estimate. Method of moments. Point estimators properties. Correctness, consistency, efficency. Bias of an estimator. • Confidence interval and interval estimate.

• Paolo Baldi, Calcolo delle probabilità e statistica 2/ed, McGraw Hill, 1998. • Alexander M. Mood, Franklin A. Graybill, Duane C. Boes, Introduzione alla statistica, McGraw-Hill, 1997. • Lecturer's notes.

• Finite and countable sample spaces. Events. Sure event, impossible event, incompatible events, jointly exhaustive events. Discrete uniformly probability theory. • Indipendent events. Conditional probability under an event. Bayes theorem. Alternatively Bayes’s law. • Basic combinatorics, permutations and combinations. Pascal triangle. Binomial expansion. • Discrete random variables. Distribution functions of discrete random variables. Discrete random vectors. Joint distribution and marginal distributions of a discrete random vector. Expected value and variance of a discrete random variable. • Indipendent discrete random variables. • Discrete random variables: uniform distribution, Bernoulli distribution, binomial distribution, geometric distribution, Poisson distribution, ipergeometric distribution. Connection between binomial and Poisson distribution. • Sigma-algebras. Kolmogorof axioms of probability theory. Borel sets. Random variables. Real valued random variables and real valued random vectors. Definition and main properties of the distribution function of a real valued random variable. Joint distribution function of a real valued random vector. • Continuos random variables. Properties of the density function of a continuous random variable. Continuous 2-dimensional random vectors. Joint distribution and marginal distributions of a continuous 2- dimensional random vector. Expected value and variance of a continuous random variable. • Indipendent continuous random variables. • Moment-generating function. • Continuous random variables: uniform continuous distribution, gamma distribution, exponential distribution, normal distribution, Chi-squared distribution. • Convergence of random variables. Almost sure convergence, convergence in probability, convergence in distribution. Weak law of large numbers. Central limit theorem. Approximate normality. • Observations and samples. Unknown parameters. Statistics. Sample mean, sample variance. Populations moments. • Statistical inference. Likelihood function. Parameters of a statistical model. Method of estimating parameters. Point estimators. Maximum likelihood estimate. Method of moments. Point estimators properties. Correctness, consistency, efficency. Bias of an estimator. • Confidence interval and interval estimate.

Classroom lectures. Classwork. The didactical matherial will be available on Teams.

Student assessment includes an oral exam on the course topics. During this exam, students must demonstrate understanding and assimilation of the course material, independent and critical analysis of the topics, identifying their most relevant aspects, and the ability to clearly and correctly present the results learned. The exam is scored using a grade out of 30. A pass grade of 18/30 is required; to achieve this score, students must clearly and correctly answer two-thirds of the questions posed, regarding the definitions, statements, and proofs covered in class. To achieve the maximum score (30/30 with honors), students must demonstrate excellent knowledge of all the topics covered in the course, correctly answer all questions, and demonstrate a critical and original approach to the subject matter.