Mathematics

GIOVANNA NAPPO
GIOVANNA NAPPO
ANTONIO AGRESTI
ANTONIO AGRESTI

General objectives: to acquire basic knowledge in probability theory.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to probability theory on finite and countable spaces, to the concept of random discrete vectors and to the concept of continuous random variable.

Applying knowledge and understanding: at the end of the course the student will be able to solve simple problems in discrete probability, problems concerning discrete random vectors and random numbers represented by continuous random variables. The student will also be able to understand the meaning and implications of independence and conditioning (in the context of discrete models), to understand the meaning of some fundamental limit theorems, such as the law of large numbers.

Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between the topics of the course with topics of mathematical analysis and combinatorics (acquired in the “Analisi I” course and treated in the course of “Fondamenti di Analisi Reale”).

Communication skills: ability to expose the contents of the course in the oral part of the test and in any theoretical questions present in the written test.

Learning skills: the acquired knowledge will allow a study, individual or given in a course related to more specialized aspects of probability theory.

General objectives: to acquire basic knowledge in probability theory.

Specific objectives:

Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to probability theory on finite and countable spaces, to the concept of random discrete vectors and to the concept of continuous random variable.

Applying knowledge and understanding: at the end of the course the student will be able to solve simple problems in discrete probability, problems concerning discrete random vectors and random numbers represented by continuous random variables. The student will also be able to understand the meaning and implications of independence and conditioning (in the context of discrete models), to understand the meaning of some fundamental limit theorems, such as the law of large numbers.

Critical and judgmental skills: the student will have the bases to analyze the analogies and the relationships between the topics of the course with topics of mathematical analysis and combinatorics (acquired in the “Calcolo” course and treated in the course of “Analisi matematica 1”).

Communication skills: ability to expose the contents of the course in the oral part of the test and in any theoretical questions present in the written test.

Learning skills: the acquired knowledge will allow a study, individual or given in a course related to more specialized aspects of probability theory.

The course requires familiarity with topics in the "Algebra Lineare" (Linear Algebra) course and the “Calcolo” (Calculus) course (sets, equivalence classes, functions, integrals, derivatives).
This knowledge is indispensable. There are no compulsory propaedeutic courses.

- Probability axioms (3h)
- Discrete probability spaces (3h)
- Combinatorics (6h)
- Inclusion-exclusion principle (4h)
- Independence (6 h)
- Binomial, multinomial and hypergeometric distributions (3h)
- Conditional probability, Product formula, Bayes formula (5 h)
- Continuity of Probability (2 h)
- Occupation numbers: Maxwell Boltzmann, Bose-Einstein, Fermi-Dirac (4h)
- Discrete random variables: expectation, variance and covariance (6h)
- Independence of random variables (2h)
- Bernoulli, binomial, geometric and negative binomial random variables (6h)
- Sums of independent random variables (2h)
- Poisson random variable and its binomial approximation (2h)
- Joint, marginal and conditional densities (4h)
- Transformation of discrete random variables (2h)
- Chebyshev's inequality and the weak law of large numbers (4h)
- Conditional expectation and its Geometric interpretation (5h)
- Continuous random variables: density function, distribution function, expectation and variance (4h)
- Uniform random variable in (0,1). Skorohod representation (3h)
- Exponential random variable as limit of Geometric random variables (2h)
- Gaussian random variables (2h)
-Transformation of continuous random variables (1h)
- De Moivre-Laplace thorem and Gaussian Approximation (2h)

NOTE Changes are possible : the other teacher will be the winner of a competion,

F. Spizzichino, G. Nappo: Introduzione al calcolo delle probabilità. (Notes will be available in the e-learning site )

Berger, Caravenna, Dai Pra: Probabilità (Springer) (for students of Sapienza University a free pdf version is available link https://link.springer.com/book/10.1007/978-88-470-4006-9)

Other notes and exercises will be available on "Sapienza" e-learning con Moodle.

S. Ross: Probabilità. (Apogeo)
A. N. Shiryaev: Probability.
W. Feller: An Introduction to Probability Theory and its Applications.

Theory (60%), Exercises (40%).

If necessary, due to sanitary rules, lessons (or part of them) will be online.

Further details, notes and exercises will be available on "Sapienza" e-learning con Moodle.

Attending lessons is not compulsary, but it is recommended

The examination aims to evaluate learning through a written test (consisting in solving problems similar to the one solved during the lectures) and an oral test (consisting in the discussion of the written examination and the most relevant topics illustrated during the course). If the student pass the written examination (with at least 18/30) he/she is admitted to the oral exam.

If necessary, due to sanitary rules, the written test (or part of it) can be substituted by an homework to be discussed during the oral examination.

To pass the exam an evaluation of not less than 18/30 is needed: this evaluation is computed on the basis of the written examination (50%) and the oral test (50%).
The student has to prove to have acquired sufficient knowledge in the topics of the program and to be able to perform at least the simplest of the assigned exercises.
To achieve a score of 30/30 “cum laude”, the student must instead demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.

See the web-page of the previous academic years 2022/23 and 2023/24
2022-23 PROBABILITA' 1 (L-Z) and 2023-24 PROBABILITA' 1 (L-Z)
LINK: https://elearning.uniroma1.it/course/view.php?id=16183 and https://elearning.uniroma1.it/course/view.php?id=17792

There will be a similar web-page for the academic year 2024/25