81624 - Probability

Learning outcomes

A working knowledge of the basics of Probability, with an eye to further applications in Stochastic Differential Equations, Brownian motions and Stochastic Processes.

Course contents

Notation and basic set theory
Sets and functions
Outer measure
Lebesgue-measurable sets and Lebesgue measure
Basic properties of Lebesgue measure
Borel sets
Lebesgue-measurable functions
Random variables
Fields generated by random variables
Probability distributions
Independence of random variables
Integral
Definition of the integral
Monotone convergence theorems
Integrable functions
The dominated convergence theorem
Relation to the Riemann integral
Approximation of measurable functions
Integration with respect to probability distributions
Absolutely continuous measures:
examples of densities
Expectation of a random variable
Characteristic function
Spaces of integrable functions
The space L
The Hilbert space L
Properties of the L -norm
Inner product spaces
Orthogonality and projections
The LP spaces: completeness
Moments
Independence
Conditional expectation (first construction)
Product measures
Independence again
Conditional probability
Strong law of large numbers
Weak convergence
Central limit theorem

Readings/Bibliography

https://www.amazon.it/Measure-Integral-Probability-Marek-Capinski/dp/1852337818/ref=sr_1_1?ie=UTF8&qid=1465801767&sr=8-1&keywords=capinski

 

https://www.amazon.it/Probability-Measure-3rd-BILLINGSLEY-PATRICK/dp/8126517719/ref=sr_1_1?ie=UTF8&qid=1465801791&sr=8-1&keywords=billingsley

Teaching methods

Regular 2 or 3 hours lectures with occasional in-course tests.

Assessment methods

Written and oral exams.

Teaching tools

Supplementary notes may be distributed during the course itself.

Office hours

See the website of Enrico Bernardi