- Docente: Pietro Rigo
- Credits: 6
- SSD: MAT/06
- Language: English
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: First cycle degree programme (L) in Statistical Sciences (cod. 8873)
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from Sep 19, 2023 to Oct 25, 2023
Learning outcomes
By the end of the course the student should know the basic theory of multidimensional random variables and sequences of random variables. In particular the student should be able: - to derive the distribution of transformed random variables - to derive joint, conditional and marginal probability density functions - to state the definition and recall the properties of multivariate normal distributions to investigate converge properties of sequences of random variables
Course contents
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Probability spaces and their elementary properties
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Various definitions of probability (just a hint)
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Random variables and their distributions
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Independence
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Probability measures on R and R^n (including distribution functions, and, in case of R, discrete, singular continuous, and absolutely continuous probability laws)
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List of the main probability measures on R and normal distribution on R^n
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Transform of random variables
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Moments
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Conditional distributions
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Convergence of random variables
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Laws of large numbers
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Central limit theorems
Readings/Bibliography
In order to prepare the exam, the notes are (more than) enough, provided obviously they are taken in a exhaustive and correct way. For those who don't like notes, and/or for those who want to deepen some topics, the following text books are suggested:
Bertsekas D.P. and Tsitsikli J.N., Introduction to Probability, 2nd Edition, ISBN: 978-1-886529-23-6
Grimmett G. and Stirzaker D. (2001) Probability and random processes, Oxford University Press.
Dall'Aglio G. (1987) Calcolo delle probabilita', Zanichelli.
Teaching methods
Regular lectures and tutorials
Assessment methods
One-hour-a-half written exam, consisting of 3/4 exercises, followed by an oral examination. The exercises are obvious versions of the ones discussed in class, and attain to elements of the syllabus covered by the course lectures. The aim of the written test is to assess the student's ability to use definitions, properties and theorems of probability theory when facing with some simple problems. Finally, the oral part of the exam is subjected to the overcoming of the written part.
Teaching tools
Notes and the text-books quoted above
Office hours
See the website of Pietro Rigo