PROBABILITY AND STATISTICS
Academic Year 2025/2026 - Teacher: Veronica BIAZZOExpected Learning Outcomes
The cours is based on the basic concepts of probability and statistics. Being a kind of introductory course, has as its objective the acquisition of the basic techniques for the interpretation in the probabilistic sense of the random type phenomena.
In particular, the course has the following objectives:
Knowledge and understanding:
Among the fundamental objectives of the course are the understanding of the statements and demonstrations of the fundamental theorems of the calculation of probabilities and statistics. The theoretical goal is to be able to build rigorous demonstrations in order to improve mathematical skills in reasoning and calculation as well as acquisition of the ability to model natural phenomena and not, to translate problems in a mathematical language in order to handle them easily and to solve them.
Applying knowledge and understanding:
Understanding the core concepts of the course has the practical aim of refining the use of logical tools and critical skills by enabling the student to deal with subjects related to the course but not performed in it.
Making judgements:
In the course, topics are proposed by comparing them with similar concepts in other subjects. It is an interest in the course to make students autonomous in the sense of improving their quality of judgment by knowing how best to deal with problems and knowing how to evaluate the correctness.
Communication skills:
The logical and application nature of the course requires and aims for clarity and lack of ambiguity in communicating.
Learning skills:
The previous goals converge in making students prepared to undergo subsequent studies with knowledge and a flexible mentality that will also be useful for incorporating the world of work.
Course Structure
The course is based on a cycle of lectures. The teacher will agree with the students of the exercises, so that they are prepared to the demands and difficulties of the written test.
Information for students with disabilities and/or SLD
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and/or dispensatory measures, based on the didactic objectives and specific needs.
Required Prerequisites
Attendance of Lessons
Detailed Course Content
1. Events and logic operations between events.
2. Setting axiomatic probability, classical definition of probability, the frequentist approach, subjective approach, the criterion of the bet, property of the probability.
3. Simple random numbers, prevision of a simple random number. Variance of a simple random number, covariance. Variance of sums and differences of random numbers, the correlation coefficient, properties, linear dependence.
4. Conditioned events and conditional probabilities.
5. Stochastic independence. Exchangeable events. Exchangeability and frequentist setting. Extractions with and without a refund from an urn of known composition, binomial and hypergeometric distribution, properties, prevision and variance. Extractions of unknown composition polls, mixtures of binomial and hypergeometric distributions. Bayes' theorem, meaning inference, likelihood values.
6. Discrete random numbers, prevision and distribution function of discrete random variables. Major distributions of discrete random variables.
7. Absolutely continuous random variables, density and distribution functions. Prevision and variance of continuous random variables. Major distributions of absolutely continuous random variables.
8. Discrete random vectors, marginal and conditional distributions, relationship between the joint distribution and marginal, stochastic independence, relationship with the incorrelation properties. Multinomial distribution.
9. Random vectors continuous, cumulative distribution function and joint probability density, marginal and conditional distributions, stochastic and incorrelation independence, probability distribution of the maximum and minimum of two random numbers, application to the case of exponential distributions. Sums of independent random variables and not, convolution integral.
10. Conditional Distributions. Generating function. Characteristic function.
11. Convergence in probability. Convergence in law. Central limit theorem.
12. Stochastic Processes. Bernoulli's process. Problem of gambler's ruin.
Textbook Information
Incertezza e Probabilita' - Scozzafava Romano - Zanichelli
Calcolo delle probabilità - Giorgio Dall'Aglio - Zanichelli
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Events and logical operations between events. Axiomatic approach to probability, classical definition of probability, frequentist approach, subjective approach, betting criterion, properties of probability. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
| 2 | Simple random numbers, prediction of a simple random number. Variance of a simple random number, covariance. Variance of sums and differences of random numbers, correlation coefficient, properties, linear dependence. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
| 3 | Conditional events and conditional probabilities, theorem of composite probabilities. Stochastic independence. Exchangeable events. Exchangeability and frequentist approach. Extractions with and without restitution from an urn of known composition, binomial and hypergeometric distribution, properties, prediction and variance. Extractions from urns of unknown composition, mixtures of binomial and hypergeometric distributions. Bayes' theorem, inferential meaning, likelihood values. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
| 4 | Discrete random numbers, prediction and distribution function of discrete random numbers. Main distributions of discrete random numbers. Absolutely continuous random numbers, probability density and distribution function. Zero probabilities, prediction and variance of continuous random numbers. Main distributions of absolutely continuous random numbers. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
| 5 | Discrete random vectors, marginal and conditional distributions, relation between joint and marginal distributions, stochastic independence, relation with the uncorrelation property. Multinomial distribution. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
| 6 | Continuous random vectors, joint probability distribution and density function, marginal and conditional distributions, stochastic independence and incorrelation, probability distribution of the maximum and minimum of two random numbers, application to the case of exponential distributions. Sums of independent and non-independent random numbers, convolution integral. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
| 7 | Conditional distributions. Generating function. Characteristic function. Convergence in probability. Convergence in law. Central limit theorem. Stochastic processes. Bernoulli process. Gambler's ruin problem. | Incertezza e Probabilita' - Scozzafava Romano - Zanichelli |
Learning Assessment
Learning Assessment Procedures
The following parameters will be taken into account in assigning the final grade:
Exam failed: the student does not possess the minimum required knowledge of the main contents of the course. The ability to use the specific language is very poor or non-existent and he is not able to independently apply the knowledge acquired.
Grade 18-21: the student has minimal knowledge of the subject, has a modest ability to integrate and critically analyze the situations presented and presents the arguments in a sufficiently clear manner although the command of language is poorly developed;
Grade 22-25: the student has a fair knowledge of the subject, even if limited to the main topics; he is able to integrate and analyze the situations presented in a critical but not always linear way and presents the topics in a fairly clear way with a fair command of language;
Grade 26-28: the student has a good knowledge of the subject, is able to integrate and analyze the situations presented in a critical and linear way, is able to solve complex problems quite autonomously and presents the topics clearly using appropriate language;
Grade 29-30 with honors: the student has an in-depth knowledge of the subject, is able to promptly and correctly integrate and critically analyze the situations presented, independently solving even highly complex problems; has excellent communication skills and command of language.