# PROBABILITY AND STATISTICS

Academic Year 2023/2024 - Teacher: Veronica BIAZZO

## Expected Learning Outcomes

The cours is based on the basic concepts of probability and statistics. Being a kind of introductory course, has as its objective the acquisition of the basic techniques for the interpretation in the probabilistic sense of the random type phenomena.

In particular, the course has the following objectives:

Knowledge and understanding:

Among the fundamental objectives of the course are the understanding of the statements and demonstrations of the fundamental theorems of the calculation of probabilities and statistics. The theoretical goal is to be able to build rigorous demonstrations in order to improve mathematical skills in reasoning and calculation as well as acquisition of the ability to model natural phenomena and not, to translate problems in a mathematical language in order to handle them easily and to solve them.

Applying knowledge and understanding:

Understanding the core concepts of the course has the practical aim of refining the use of logical tools and critical skills by enabling the student to deal with subjects related to the course but not performed in it.

Making judgements:

In the course, topics are proposed by comparing them with similar concepts in other subjects. It is an interest in the course to make students autonomous in the sense of improving their quality of judgment by knowing how best to deal with problems and knowing how to evaluate the correctness.

Communication skills:

The logical and application nature of the course requires and aims for clarity and lack of ambiguity in communicating.

Learning skills:

The previous goals converge in making students prepared to undergo subsequent studies with knowledge and a flexible mentality that will also be useful for incorporating the world of work.

## Course Structure

The course is based on a cycle of lectures. The teacher will agree with the students of the exercises, so that they are prepared to the demands and difficulties of the written test.

## Required Prerequisites

The knowledge of the concepts and techniques concerning the courses of Mathematical Analysis is fundamental. In particular, knowledge of simple integrals, multiple and series is essential.

## Attendance of Lessons

Strongly recommended.

## Detailed Course Content

1. Events and logic operations between events.

2. Setting axiomatic probability, classical definition of probability, the frequentist approach, subjective approach, the criterion of the bet, property of the probability.

3. Simple random numbers, prevision of a simple random number. Variance of a simple random number, covariance. Variance of sums and differences of random numbers, the correlation coefficient, properties, linear dependence.

4. Conditioned  events and conditional probabilities.

5. Stochastic independence. Exchangeable events. Exchangeability and frequentist setting. Extractions with and without a refund from an urn of known composition, binomial and hypergeometric distribution, properties, prevision and variance. Extractions of unknown composition polls, mixtures of binomial and hypergeometric distributions. Bayes' theorem, meaning inference, likelihood values.

6. Discrete random numbers, prevision and distribution function of discrete random variables. Major distributions of discrete random variables.

7. Absolutely continuous random variables, density and distribution functions. Prevision and variance of continuous random variables. Major distributions of absolutely continuous random variables.

8. Discrete random vectors, marginal and conditional distributions, relationship between the joint distribution and marginal, stochastic independence, relationship with the incorrelation properties. Multinomial distribution.

9. Random vectors continuous, cumulative distribution function and joint probability density, marginal and conditional distributions, stochastic and incorrelation independence, probability distribution of the maximum and minimum of two random numbers, application to the case of exponential distributions. Sums of independent random variables and not, convolution integral.

10. Conditional Distributions. Generating function. Characteristic function.

11. Convergence in probability. Convergence in law. Central limit theorem.

12. Stochastic Processes. Bernoulli's process. Problem of gambler's ruin.

## Course Planning

SubjectsText References
1 Eventi ed operazioni logiche tra eventi. Impostazione assiomatica della probabilità, definizione classica della probabilità, impostazione frequentista, impostazione soggettiva, criterio della scommessa, proprieta' della probabilita'.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
2Numeri aleatori semplici, previsione di un numero aleatorio semplice. Varianza di un numero aleatorio semplice, covarianza. Varianza di somme e differenze di numeri aleatori, coefficiente di correlazione, proprietà, dipendenza lineare.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
3Eventi condizionati e probabilità condizionate, teorema delle' probabilita' composte.Indipendenza stocastica. Eventi scambiabili. Scambiabilita' e impostazione frequentista. Estrazioni con e senza restituzione da un’urna di composizione nota, distribuzione binomiale e ipergeometrica, proprietà, previsione e varianza. Estrazioni da urne di composizione incognita, misture di distribuzioni binomiali e ipergeometriche. Teorema di Bayes, significato inferenziale, valori di verosimiglianza.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
4Numeri aleatori discreti, previsione e funzione di ripartizione di numeri aleatori discreti. Principali distribuzioni di numeri aleatori discreti.Numeri aleatori assolutamente continui, densita' di probabilita' e funzione di ripartizione. Probabilita' nulle, previsione e varianza di numeri aleatori continui. Principali distribuzioni di numeri aleatori assolutamente continui.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
5Vettori aleatori discreti, distribuzioni marginali e condizionate, relazione tra la distribuzione congiunta e le marginali, indipendenza stocastica, relazione con la proprietà di incorrelazione. Distribuzione multinomiale.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
6Vettori aleatori continui, funzione di ripartizione e densità di probabilità congiunta, distribuzioni marginali e condizionate, indipendenza stocastica e incorrelazione, distribuzione di probabilità del massimo e del minimo di due numeri aleatori, applicazione al caso di distribuzioni esponenziali. Somme di numeri aleatori indipendenti e non, integrale di convoluzione.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
7Distribuzioni condizionate. Funzione generatrice. Funzione caratteristica. Convergenza in probabilita'. Convergenza in legge. Teorema del limite centrale. Processi stocastici. Processo di Bernoulli. Problema della rovina del giocatore.Incertezza e Probabilita' - Scozzafava Romano - Zanichelli

## Learning Assessment

### Learning Assessment Procedures

The evaluation method consists of a written and an oral test. The written test is useful to understand if the ability (required in the course objectives) has been reached in modeling phenomena in a rigorous way. The minimum mark for the written exam to have access to the oral exam is 15/30. The grade of the written exam strongly influences the final grade. The oral exam is useful for understanding the quality of theoretical knowledge of the subject and for evaluating the ability to know how to build a rigorous demonstration. Verification of learning can also be carried out electronically, should the conditions require it.

### Examples of frequently asked questions and / or exercises

In Stadium you can find several exercises.