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

• Probability spaces and Kolmogorov axioms
  
• Conditional probability, Law of Total Probability and Bayes' formula
• Independent events
  
• Random variables and distribution functions, discrete and continuous random variables, expected value, variance and covariance
  
• Discrete models: Bernoulli, Binomial, Hypergeometric, Poisson, Geometric
• Continuous models: Uniform, Gaussian, Gamma, Student, Fisher.
  
• Law of Large Numbers and applications
  
• Central Limit Theorem and applications

Readings/Bibliography

Alberto Lanconelli, Introduction to Probability Theory (2023) ISBN-13: ‎979-8850457037

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. In case of online exam, this will be supported by the softwares Teams, Zoom and EOL (https://eol.unibo.it/)

Teaching tools

Exercises with solutions

Office hours

See the website of Alberto Lanconelli