Probability and Statistics
A.Y. 2024/2025
Learning objectives
The course is meant to provide the student with a deep knowledge of probability theory and with the statistical methods on which physics and data modelling are grounded. Besides giving the definitions and the concepts of frequentist and Bayesian interpretation of probability, operative tools will be discussed, such as the law of large numbers, the central limit theorem, the use of generating functions and the principle of maximum entropy.
Expected learning outcomes
The student will be able to use the concepts of probability theory to analyse data, make statistical inference and statistical tests. Moreover, they will be confident with the theoretical bases of statistical mechanics and with its tools.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Elements of probability theory:
The definition of probability: frequent and Bayesian approach
Conditional probability
Average, moments, standard deviation, generating function
Random discrete variables
- Combinatorics calculation
- Binomial and multinomial distributions
- Occupation numbers
Limits.
- Law of large numbers
- Poisson processes
Continuous distributions
- probability density
- cumulated distributions
- characteristic functions
- examples: Uniform, Gaussian, Exponential, Beta, Gamma, etc.
Extreme-value and rank statistics
Sum statistics. Central limit, stable distributions
Principle of maximum entropy
Elements of random matrix theory
Statistical estimates:
Confidence intervals
Maximum likelihood
Regression
Statistical tests:
The design of an experiment
statistical significance
P-value: T-test, F-test, KS-test
Statistical visualization:
Univariate distributions: histograms, boxplot, swarmplot
Multivariate distributions: scatter/line plot, density plot, grids
Dimensional reduction
Networks
The definition of probability: frequent and Bayesian approach
Conditional probability
Average, moments, standard deviation, generating function
Random discrete variables
- Combinatorics calculation
- Binomial and multinomial distributions
- Occupation numbers
Limits.
- Law of large numbers
- Poisson processes
Continuous distributions
- probability density
- cumulated distributions
- characteristic functions
- examples: Uniform, Gaussian, Exponential, Beta, Gamma, etc.
Extreme-value and rank statistics
Sum statistics. Central limit, stable distributions
Principle of maximum entropy
Elements of random matrix theory
Statistical estimates:
Confidence intervals
Maximum likelihood
Regression
Statistical tests:
The design of an experiment
statistical significance
P-value: T-test, F-test, KS-test
Statistical visualization:
Univariate distributions: histograms, boxplot, swarmplot
Multivariate distributions: scatter/line plot, density plot, grids
Dimensional reduction
Networks
Prerequisites for admission
Basic knowledge of calculus
Teaching methods
The course will combine face-to-face lessons with classroom exercises in small groups under the supervision of the teacher. Part of the course will include practical methods for statistical analysis in python.
Teaching Resources
Von der Linden, Wolfgang, Volker Dose, and Udo Von Toussaint. Bayesian probability theory: applications in the physical sciences. Cambridge University Press, 2014.
Bohm, Gerhard, and Günter Zech. Introduction to statistics and data analysis for physicists. Vol. 1. Hamburg: Desy, 2010.
https://s3.cern.ch/inspire-prod-files-d/da9d786a06bf64d703e5c6665929ca01
Bohm, Gerhard, and Günter Zech. Introduction to statistics and data analysis for physicists. Vol. 1. Hamburg: Desy, 2010.
https://s3.cern.ch/inspire-prod-files-d/da9d786a06bf64d703e5c6665929ca01
Assessment methods and Criteria
The examination consists in an excercise to be done within one week. A dataset will have to be analyzed and a python notebook should be produced. The notebook will be discussed in an oral exam together with the topics covered in the course.
FIS/03 - PHYSICS OF MATTER - University credits: 3
FIS/04 - NUCLEAR AND SUBNUCLEAR PHYSICS - University credits: 3
FIS/04 - NUCLEAR AND SUBNUCLEAR PHYSICS - University credits: 3
Lessons: 42 hours
Professor:
Zapperi Stefano
Educational website(s)
Professor(s)
Reception:
11-12 Wednesday
office at the physics department or zoom (send email for an appointment)