04642 - Probability Calculus and Statistics

Learning outcomes

At the end of the course, the student knows basic concepts and methods of probability and mathematical statistics. The student can solve simple problems of probability and statistical inference.

Course contents

○ Mathematical model of a random experiment: sample space, events; axiomatic definition of probability and its properties.

○ Classical definition of probability and combinatorics.

○ Conditional probability and independence: formula of total probability and Bayes' theorem.

○ Random variables

• Law (or distribution) and cumulative distribution function.
• Discrete random variables and (absolutely) continuous random variables: discrete density and continuous density.
• Expected value and variance.
• Common probability distributions: Bernoulli, binomial, Poisson, discrete uniform, continuous uniform, exponential, normal (or Gaussian).

○ Random generators

○ Random vectors

• Joint law, marginal laws, joint cumulative distribution function, independence of random variables, covariance.
• Discrete random vectors: joint discrete density and marginal densities.

○ Limit theorems

• Sequence of i.i.d. random variables.
• Law of large numbers: Chebyshev's inequality, Monte Carlo method.
• Central limit theorem.

○ Discrete-time Markov chains: transition matrix, directed graph representation, n-step transition probability, communication classes, invariant distribution.

Readings/Bibliography

Lectures notes and sheets of exercises prepared by the teacher (available on "Insegnamenti OnLine").

Supplementary textbook: Paolo Baldi, Introduzione alla probabilità con elementi di statistica, seconda edizione, McGraw-Hill.

 

Teaching methods

Lectures and exercises will be alternated in order to explain theoretical concepts through examples.

Lectures are not compulsory but are strongly recommended, as active participation helps to obtain a deeper comprehension of the topics treated in the course.

Assessment methods

The assessment is based on a written exam, which lasts approximately two hours, and consists in 4/6 exercises and 1/2 theoretical questions.

The exercises of the exam will be of the same type of the exercises proposed on the "sheets of exercises" (available on "Insegnamenti OnLine").

The theoretical questions concern the more "theoretical" topics of the course, which will be listed on "Insegnamenti OnLine".

Teaching tools

Website of the course available on "Insegnamenti OnLine", where the student can find: lecture notes, sheets of exercises, past written exams, and other useful material for the course.

Office hours

See the website of Andrea Cosso