# 98721 - INTRODUCTION TO PROBABILITY AND THE MATHS OF RISK

• Docente: Paolo Guasoni
• Credits: 6
• SSD: MAT/06
• Language: English
• Teaching Mode: Blended Learning
• Campus: Bologna
• Corso: Second cycle degree programme (LM) in Greening Energy Market and Finance (cod. 5885)
• from Sep 20, 2023 to Oct 31, 2023

## Learning outcomes

At the end of the course the student has good knowledge of probability theory of discrete and continuous random variables. Particular attention is paid to the theory of stochastic processes, both diffusive and with jumps. The student masters the main techniques of stochastic calculus applied to finance, such as stochastic differential and integral domain and change of measure techniques.

## Course contents

This course is a problem-based introduction to probability and stochastic processes. No previous knowledge of probability is assumed, but knowledge of calculus in one or more variables is required.

1. Axiomatic definition of probability. Uniform probability spaces. Counting methods: replacement, ordering. Conditional probability. Independence for events. The law of total probability. Bayes' rule.
2. Discrete random variables. Independence for random variables. Joint, marginal, and conditional densities. Common random variables and their interpretation: Bernoulli, dicrete uniform, binomial, hypergeometric, geometric, Poisson, Pascal.
3. Expectation of dicrete random variables. Variance and its properties. Expectation and variance of common random variables. Covariance and correlation. Variance of a sum. Null correlation and independence. Linear prediction.
4. Conditional expectation and its properties. Conditional Variance. Sigma-algebras, Continuous Random variables. The Uniform and Exponential distributions. Distribution functions and densities.
5. Marginal, joint and conditional densities. Gamma, Normal and Cauchy distribution. Derived Distributions: monotonic and general case. Conditional Expectation. Law of total expectation. Markov and Chebishev Inequalities.
6. Convergence of Random Variables. The Weak and Strong Laws of Large Numbers. Characteristic Functions and their properties. CF of a sum. CF of common random variables. The Central Limit Theorem.

Introduction to Probability, 2nd edition Dimitri P. Bertsekas, John N. Tsitsiklis Athena Scientific ISBN: 1886529236

Weighing the Odds: A Course in Probability and Statistics David Williams Cambridge University Press ISBN: 052100618X

## Teaching methods

The course includes weekly lectures and problem sessions. Some problem sessions may take place online.

## Assessment methods

The course assessment consists in a two-hour written final exam, which requires students to solve a few problems. Students need to understand how to translate the problems into probability models, how to solve the models, and explain the meaning of the solution for the original problems. The exam's valuation is based on the number of problem solved and on the completeness and accuracy of their solutions. The are no midterms.

## Teaching tools

Textbook, lecture slides, problems. The course participates in the university's teaching enhancement project.

## Office hours

See the website of Paolo Guasoni