KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will know the elementary theory of probability. In particular, the concepts of conditional probability and random variable.

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 JUDGMENTS
The student will autonomously recognize the probabilistic environment of a problem and will know how to apply the most suitable models and
methods.

COMMUNICATIVE SKILLS
At the end of the course the student will be able to express simple but precise concept in this discipline.

LEARNING SKILLS
At the end of the course the student will be able to consult medium-level university textbook in probability.

Knowledge of basic mathematics

Preliminary Concepts
Fundamental properties of probability. Conditional probability. Independent events. Decomposition formula,
law of total probability, Bayes' rule. Independent trials.

Random Variables - Part One
The general notion of a random variable. Discrete real-valued random variables. Expectation of a discrete random variable and generalization to any random variable. Variance and covariance, their properties. Independent random variables. Chebyshev's inequality. Extension of the theory to vector-valued random variables. Notable distributions: binomial, hypergeometric, geometric, Poisson.

Random Variables - Part Two
Absolutely continuous random variables. Density of a random variable, distribution function. Expectation of an absolutely continuous random variable. Notable distributions: uniform, exponential, Gamma, Gaussian.

Random Variables - Part Three
Sample mean. The law of large numbers. Central limit theorem. Chi-square and Student's t distributions.

Statistics
Descriptive statistics. Inferential statistics: statistical models, samples, and estimators. Methods for finding point estimators: maximum likelihood. Properties of point estimators: unbiasedness and consistency. Confidence intervals and interval estimation. Sampling from the normal distribution: confidence intervals for the mean and the variance. Hypothesis testing. Output of a test, Type and Type II errors. Decision procedure. Tests for Gaussian samples.

Berger, Caravenna e Dai Pra - Probabilità, un primo corso attraverso esempi, modelli e applicazioni - Springer Milan (2021)

Preliminary Concepts

Fundamental properties of probability. Conditional probability. Independent events. Decomposition formula,
law of total probability, Bayes' rule. Independent trials.

Random Variables - Part One
The general notion of a random variable. Discrete real-valued random variables. Expectation of a discrete random variable and generalization to any random variable. Variance and covariance, their properties. Independent random variables. Chebyshev's inequality. Extension of the theory to vector-valued random variables. Notable distributions: binomial, hypergeometric, geometric, Poisson.

Random Variables - Part Two
Absolutely continuous random variables. Density of a random variable, distribution function. Expectation of an absolutely continuous random variable. Notable distributions: uniform, exponential, Gamma, Gaussian.

Random Variables - Part Three
Sample mean. The law of large numbers. Central limit theorem. Chi-square and Student's t distributions.

Statistics
Descriptive statistics. Inferential statistics: statistical models, samples, and estimators. Methods for finding point estimators: maximum likelihood. Properties of point estimators: unbiasedness and consistency. Confidence intervals and interval estimation. Sampling from the normal distribution: confidence intervals for the mean and the variance. Hypothesis testing. Output of a test, Type and Type II errors. Decision procedure. Tests for Gaussian samples.

Lectures, both of theoretical character and focusing on the solution of
exercises

The assessment includes a written test and a subsequent oral test. The written test consists of probability and statistics exercises, as well as theoretical questions. Specifically, there are 5 exercises worth 3 points each and 15 theory questions worth 1 point each. The written test score, out of thirty, is the sum of the scores obtained in the individual exercises and questions. The student is admitted to the oral exam if they have obtained at least the minimum score of 18/30 in the written test. The oral exam questions focus on the definition of basic concepts, the proofs of the theorems presented in the course, and the reasoned understanding of the studied concepts. The assessment will consider the degree of knowledge, the accuracy of the exposition, and the ability to articulate independent reflections. The oral exam is assigned a score out of thirty (independent of the score obtained in the written exam). The final grade averages the score obtained in the written exam with the evaluation received in the oral exam (which must still be at least 18/30). Honors are awarded to those who have performed particularly well and have achieved the maximum score in both the written and oral exams.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs)