79222 - Probability I

Learning outcomes

By the end of the course module the student should know the basic tools of probability calculus, with a special focus on their role in the statistical analysis. In particular, the student should be able to: - compute the probability of events, by using the axioms and the fundamental theorems of probability calculus - identify the main discrete and continuous random variables and compute their expected values and variances - analytically treat univariate and bivarate random variables.

Course contents

• Discrete probability spaces, conditional probability, law of total probability, Bayes' formula, independent events. Discrete random variables, probability function, joint probability function, independent random variables, expected value, variance, covariance. Models of discrete random variables: Bernoulli, Binomial, Poisson, Geometric.
• General probability spaces, random variables, distribution function, independent random variables, continuous random variables, probability density function, expected value, variance and covariance for continuous random variables. Models of continuous random variables: Uniform, Gaussian, Gamma, Student and Fisher.
  
• Law of large numbers and Central Limit theorem.

Readings/Bibliography

Lecture notes. Suggested readings:

• Introduction to Probability, 2nd Edition, by Dimitri P. Bertsekas and John N. Tsitsikli, ISBN: 978-1-886529-23-6

Teaching methods

Regular lectures and tutorials

Assessment methods

One-hour written exam, articulated in a series of 2 exercises each with a maximum grade of 15 points, followed by an oral examination. The written test is aimed at assessing the student's ability to use the definitions, properties and theorems of probability theory in solving theoretical exercises. Every exercise attains to elements of the syllabus covered during the course lectures. Online exams will be supported by the softwares Teams, Zoom and EOL (https://eol.unibo.it/)

Teaching tools

Slides and exercises with solutions

Office hours

See the website of Alberto Lanconelli