INGLESE

Elements of measure theory in R, with special concern with the integral representation of measures. Elements of Theory of Distributions. Convex functions. Finite Difference Equations. Special properties of the expected value. Conditional expected value. Special functions associated with a random variable and its moments. Uniform process. The Poisson process. The random walk. The gambler's ruin. Discrete Markov chains. Martingales.

Elements of measure theory in R, with special concern with the integral representation of measures. Elements of Theory of Distributions. Convex functions. Finite Difference Equations. Special properties of the expected value. Conditional expected value. Special functions associated with a random variable and its moments. Uniform process. The Poisson process. The random walk. The gambler's ruin. Discrete Markov chains. Martingales.

Acquisizione della capacità di comprendere i metodi del calcolo delle probabilità e del loro utilizzo in vista delle applicazioni statistiche

Una buona conoscenza dell'algebra e almeno dell'analisi matematica delle funzioni di una variabile

Lezioni frontali, con libera discussione di numerosi esempi e problemi

Esame orale, che trae spunto dalla discussione di un problema applicativo e dalla sua soluzione per l'analisi del possesso delle nozioni teoriche fondamentali

Calcolo Combinatorio: il fattoriale, disposizioni semplici e con ripetizioni, permutazioni, combinazioni semplici, coefficienti binomiali.

Probabilità elementare, definizione classica, frequenza relativa e probabilità, proprietà assiomatiche della probabilità; esempi ed esercizi.

Probabilità condizionata. Dipendenza e indipendenza di eventi. Teoremi fondamentali: doppio condizionamento, moltiplicazione, legge delle alternative. Legge di Bayes.
Esempi ed applicazioni.

La nozione di variabile aleatoria. Variabili aleatorie discrete. Valore atteso. Varianza. Funzione di una variabile aleatoria. Variabili aleatorie congiunte, indipendenti e dipendenti. Somma di variabili aleatorie. Prodotto di variabili aleatorie.
Teoremi fondamentali su valori attesi e varianze. Variabili aleatorie standardizzate. Diseguaglianza di Chebychev. Leggi dei grandi numeri.

Variabili aleatorie speciali: variabile dicotomica, variabile uniforme, variabile binomiale, variabile geometrica, variabile binomiale negativa, variabile di Poisson. Distribuzione di Gauss.

Teorema Centrale del Limite.

English

Elements of measure theory in R, with special concern with the integral representation of measures. Elements of Theory of Distributions. Convex functions. Finite Difference Equations. Special properties of the expected value. Conditional expected value. Special functions associated with a random variable and its moments. Uniform process. The Poisson process. The random walk. The gambler's ruin. Discrete Markov chains. Martingales.

B. V. GNEDENKO & A. YA. KHINCHIN, An elementary introduction to the theory of probability, W. H. Freeman and Company, San Francisco/London (1961)

Y. A. ROZANOV, Probability Theory: a concise course,Dover, New York (1969)

Lecture notes

Acquisition of the ability to understand the methods for the computation of probabilities in many different contexts, and their use in view of statistical applications

A good acquaintance with elementary algebra and of Calculus, at least as far as the functions of one variable are concerned

Front lectures and free discussions of several examples and problems

Oral examination, starting from the discussion of a problem arising from applications to arrive at the analysis of student's acquaintance with basic theoretical notions

Combinatorial Calculus: factorials, dispositions (simple and with repetitions, permutations, simple choices, binomial coefficients.

Elementary probability, classical definition, relative frequency and probability, axiomatic properties of probability; examples and exercises.

Conditional probability. Dependence and independence of events. Basic theorems: double conditioning, multiplication, law of alternatives. Bayes's law. Examples and applications.

The notion of a random variable. Discrete random variables. Expected value. Variance. Functions of a random variable. Joint random variables, independent and dependent. Sum of random variables. Product of random variables.
Basic theorems on expected values and variances. Standard random variables. Chebychev's inequality. Laws of large numbers.

Special random variables: dichotomic random variable, uniform random variable, binomial random variable, geometric random variable, variabile negative binomial random variable, Poisson's random variable. Gauss's distribution.

Central Limit Theorem.