# ELEMENTS OF PROBABILITY

## Learning outcomes of the course unit

Know, understand and be able to explain all the essential arguments in the section "Programma esteso" below, which form an essential background of probability and statistics for the applications

Be able to solve exercises and problems on the course arguments, in particular all the "homeworks" assigned during the lessons and all the exercises of the book [Ross] from chapters 3-8

Be able to check whether a phenomenon is non deterministic and when it is possible to model it with one of the standard models of random variables presented

Be able to read and understand scientific texts which build on the knowledge of inferential statistics in one variable

It is expected that the following relevant Dublin descriptors are met:

The bachelor graduate:

- possesses excellent knowledge on one variable Calculus

- is able to read and understand advanced textbooks and research papers in mathematics.

The bachelor graduate:

- is able to create rigorous proofs of mathematical statements similar to ones seen already

- is able to translate problems in a mathmatical formulation to analyze and solve them

- is able to understand possible links between different fields of mathematics and with other subjects

The bachelor graduate:

- is able to develop logical arguments, given hypothesis and thesis

- is able to identify correct proofs and to detect fallacious reasoning

- is able to adapt simple mathematical models taken from other subjects and use them to ease the solution of a problem

The bachelor graduate:

- is able to communicate problems, ideas and solutions about mathematics both to experts and non-experts, both in his/her language and in English, in written and oral form

- has team-work experience but is also able to work alone

The bachelor graduate:

- is able to continue its studies with a high degree of autonomy both in Mathematics and in other scientific disciplines

- has a flexible mentality and is able to quickly fit in work enviroments, with the ability to confront new problems

- is capable to acquire new knowledge in mathematics or about his/her job activity by individual study

- is able to recover with ease information from scientific literature

The aim of the course consists in providing students with the basic knowledges of Probability theory and Measure theory.

## Prerequisites

Analisi matematica 1

## Course contents summary

Combinatorics. Elementary probability theory. Discrete and continuous models of random variables. Inferential statistics in one variable. Confidence intervals. Classical statistical tests.

1. Some elements of combinatorics.

2. Axioms of probability.

3. Conditional probability and independence.

4. Probability on a countable space.

5. Some topics from measure theory.

Exterior measures. Construction of a measure. Caratheodory theorem. Lebesgue measure. Main properties of positive measures. Measurable functions/randon variables. Integrable functions. Monotone convergence theorem, Fatou's lemma, dominated convergence theorem. L^p spaces. L^2 viewed as an Hilbert space.

6. Independent random variables (r.v.).

7. Probability distribution on R.

8. Probability distributions on R^n.

9. Characteristic functions and their properties.

10. Sums of independent random variables.

11. Gaussian r.v.

12. Convergence of r.v. (convergence in probability, weak convergence)

13. The law of large numbers.

14. The central limit theorem.

15. Conditional expectation.

16 Martingales, sub- and supermartingales.

## Course contents

Sample space, events and their composition, axioms of probability. Finite spaces with equiprobable elementary events, combinatorics. Counting functions, injective functions (dispositions), permutations, combinations (binomial coefficient). Conditional probability, total probability formula, Byas formula. Binary tests, sensitivity, specificity and paradoxes. Independent events.

Discrete random variables, probability mass function, cumulative distribution function. Continuous random variables, probability density function, cumulative distribution function. Expected value, expectation of a function, linearity. Variance, standard deviation, mode, median, quartiles, range. Linear and non-linear transformations.

Random vectors, joint law and marginal laws, independence, covariance, correlation. Sum, min and max of random variables.

Models of discrete random variables: uniform, Bernoulli, binomial, Poisson, geometric, hypergemetric.

Models of continuous random variables: uniform, exponential, Gaussian.

Law of the sum of several independent random variables, reproducibility, momoent generating function, weak law of large numbers, central limit theorem.

Randomness in the industrial processes, process control. Accuracy, precision and capability of a process.

Inferential statistics, population, sample, sample statistics, unbiased and consistent estimators. Sample mean, sample variance, distribution in the Gaussian case, chi-squared law.

Confidence intervals, bilateral and unilateral. Auxiliary functions: Gaussian case, estimation of the mean with variance known or unknown, estimation of the variance with mean known or unknown; exponential case, estimation of the parameter; Bernoulli case, estimation of p; two Gaussian samples, estimation of the difference of the means (homoskedastic hypothesis), estimation of the ratio of the variances.

Classical hypothesis testing, bilateral and unilateral. Test performed on the statistics, on the estimator and on the p-value. Operating characteristic curve, industrial language.

1. Some elements of combinatorics.

2. Axioms of probability.

3. Conditional probability and independence.

4. Probability on a countable space.

5. Some topics from measure theory.

Exterior measures. Construction of a measure. Caratheodory theorem. Lebesgue measure. Main properties of positive measures. Measurable functions/randon variables. Integrable functions. Monotone convergence theorem, Fatou's lemma, dominated convergence theorem. L^p spaces. L^2 viewed as an Hilbert space.

6. Independent random variables (r.v.).

7. Probability distribution on R.

8. Probability distributions on R^n.

9. Characteristic functions and their properties.

10. Sums of independent random variables.

11. Gaussian r.v.

12. Convergence of r.v. (convergence in probability, weak convergence)

13. The law of large numbers.

14. The central limit theorem.

15. Conditional expectation.

16 Martingales, sub- and supermartingales.

## Recommended readings

Francesco Morandin - Note dell'insegnamento 2018 (redatte man mano e disponibili online dopo ogni lezione)

Sheldon Ross - Introduction to probability and statistics for engineers and scientists

J. Jacob, P. Protter: Probability essentials. Springer-Verlag, Berlino 2000.

D. Williams, Probability with martingales, Cambridge mathematical textbook, Cambridge University Press 1991.

## Teaching methods

Traditional classes and exercise sessions. Arguments are presented in a practical way and formalized only when useful. Much stress is given to the motivations and many examples are presented. Applied exercises and theoretical homework (the latter are optional) are assigned regularly during lessons. During exercise sessions will be presented the solution of some of the exercises and problems assigned in the previous lessons.

Lectures and classroom exercises.

During the lectures the main results from measure theory and from probability theory are discussed and for almost all of them complete proofs are provided. Some examples of applications of such results are provided. The classroom exercises are aimed at showing with more details how and where the abstract results can be applied to make students understand better the relevance of what they are studying.

## Assessment methods and criteria

The examination is a written test with two probability problems and two statistics problems. The student is required to solve three out of four, in fact all four problems will be graded, but only the best three scores will be added to grade the whole test. (Each problem is worth 11-13 points and it is split in three parts, the first one is quite elementary and worth about 7 points, the other two are more advanced and worth 2-3 points each.) The final score of the test is given by the sum of the three problems with the highest scores, possibly increased by a percentage bonus for early hand in of the exam paper. There is an upper bound to 30 points, with ""laude"" given if the total was at least 33.

It is possible to ask for an oral examination after the written one, but if the final evaluation is not positive, the student must redo the written part.

To pass the exam the student should master the mathematical language and formalism. He must know the mathematical objects and the theoretical results of the course and he should be able to use them with ease.

The exam consists of two parts: a written part and an oral part. The written part is based on some exercises and it is aimed at evaluating the skills of the student in applying the abstract results proposed during the course to some concrete situations. The oral part is aimed at evaluating the knowledge of the abstract results seen during the course and their proofs.

## Other informations

Il materiale didattico disponibile sul sito di e-learning dell'insegnamento comprende i video e le lavagnate delle lezioni, che sono svolte tramite tablet computer