Teacher

Marco Minozzo

Referent

Marco Minozzo

Credits

7.5

Also offered in courses:

• Introduction to Probability (module I) of the course Scuola di Dottorato

Language

English

Class attendance

Free Choice

Location

VERONA

Moodle Seminars 0

Learning objectives

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

Prerequisites and basic notions

Basic Calculus and basic knowledge of probability theory. In particular, students should have been exposed to the material in Lectures 1, 2, 3, 4, 5, 6, 8 of the MIT online course “Introduction to Probability” (RES.6-012) by John Tsitsiklis and Patrick Jaillet
https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/
Attendance to more advanced courses such as real analysis, probability, distribution theory and statistical inference would be desirable.

Program

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.

Bibliography

Didactic methods

The lesson will be delivered in presence. Topics will be illustrated with the use of many examples.

Learning assessment procedures

A two-hour written paper at the end of the course. No material is permitted during the examination.

Scheduled Lessons