# Probability (2020/2021) Codice insegnamento
4S003795
Docente
Marco Minozzo
Coordinatore
Marco Minozzo
crediti
7,5
Settore disciplinare
SECS-S/01 - STATISTICA
Lingua di erogazione
Italiano
Sede
VERONA
Periodo
A.A. 20/21 dottorato dal 1-ott-2020 al 30-set-2021.

#### Obiettivi formativi

Availability

The course is intended for 1st year students on PhD in Economics and Management.

Pre-requisites

Introduction to mathematics, elementary statistical theory and elementary set theory. Basic knowledge of probability theory, as in: P. Newbold, W. Carlson, B. Thorne (2012), Statistics for Business and Economics, Pearson Higher Education, Chapters 3-5 (previous editions would be fine as well). Attendance at more advanced courses such as real analysis, probability, distribution theory and statistical inference would be desirable.

Objectives of the course

The purposes of this course are: (i) to explain, at an intermediate level, the basis of probability theory and some of its more relevant theoretical features; (ii) to explore those aspects of the theory most used in advanced analytical models in economics and finance. The topics will be illustrated and explained through many examples.

#### Programma

Course content

1. Algebras and sigma-algebras, axiomatic definition of probability, probability spaces, properties of probability, conditional probability, Bayes theorem, stochastic independence for events.
2. Random variables, measurability, cumulative distribution functions and density functions.
3. Transformations of random variables, probability integral transform.
4. Lebesgue integral, expectation and variance of random variables, Markov inequality, Tchebycheff inequality, Jensen inequality, moments and moment generating function.
5. Multidimensional random variables, joint distributions, marginal and conditional distributions, stochastic independence for random variables, covariance and correlation, Cauchy-Schwartz inequality.
6. Bivariate normal distribution, moments, marginal and conditional densities.
7. Transformations of multidimensional random variables.
8. Convergence of sequences of random variables, weak law of large numbers and central limit theorem.

Textbook

S. Ross (2010). A First Course in Probability, 8th Edition. Pearson Prentice Hall.

G. Casella, R. L. Berger (2002). Statistical Inference, Second edition. Duxbury Thompson Learning.
R. Durrett (2009). Elementary Probability for Applications. Cambridge University Press.
M. J. Evans, J. S. Rosenthal (2003). Probability and Statistics - The Science of Uncertainty. W. H. Freeman and Co.
G. Grimmett, D. Stirzaker (2001). Probability and Random Processes. Oxford University Press.
A. M. Mood, F. A. Graybill, D. C. Boes (1974). Introduction to the Theory of Statistics. McGraw-Hill.
P. Newbold, W. Carlson, B. Thorne (2012). Statistics for Business and Economics. Pearson Higher Education.
D. Stirzaker (2003). Elementary Probability. Cambridge University Press.
L. Wasserman (2004). All of Statistics. Springer.

R. B. Ash, C. A. Doléans-Dade (2000). Probability and Measure Theory. Harcourt/Academic Press.
M. J. Schervish (1995). Theory of Statistics. Springer.

 Autore Titolo Casa editrice Anno ISBN Note S. Ross A First Course in Probability (Edizione 8) Pearson Prentice Hall 2010 L. Wasserman All of Statistics Springer 2004 D. Stirzaker Elementary Probability Cambridge University Press 2003 R. Durrett Elementary Probability for Applications Cambridge University Press 2009 A. M. Mood, F. A. Graybill, D. C. Boes Introduction to the Theory of Statistics McGraw-Hill 1974 R. B. Ash, C. A. Doléans-Dade Probability and Measure Theory Harcourt/Academic Press 2000 G. R. Grimmett, D. R. Stirzaker Probability and Random Processes (Edizione 3) Oxford University Press 2001 0198572220 M. J. Evans, J. S. Rosenthal Probability and Statistics - The Science of Uncertainty W. H. Freeman and Co. 2003 G. Casella, R. L. Berger Statistical Inference (Edizione 2) Duxbury Thompson Learning 2002 P. Newbold, W. Carlson, B. Thorne Statistics for Business and Economics Pearson Higher Education 2012 M. J. Schervish Theory of Statistics Springer 1995

#### Modalità d'esame

A two-hour written paper at the end of the course.