Probability Lab
A.Y. 2025/2026
Learning objectives
The main objective of the course is to make students aware of the fundamental tools for the study of probabilistic models for the counting of successes both in the discrete and continuous time, including the tools for simulating random experiments. Simple applications are presented by means of the study of a specific class of stochastic processes: Markov chains. The concept of the laboratory is "learning by doing": each student is guided towards the know-how by means of a computational work: simulation of random variables and fundamental stochastic processes. Mathematical software, as Matlab and/or R are the computational instruments.
Expected learning outcomes
The student will be able to recognize the main probabilistic models for the counting processes, simulate them and recognize their main properties by means of the samples. The student will be able to deal with simple stochastic processes. Awareness about how theoretical mathematical properties are the foundation of the simulating algorithms is expected, by means of software as Matlab and/or R.
Lesson period: First semester
Assessment methods: Giudizio di approvazione
Assessment result: superato/non superato
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. The Dice Lab:
1.1. Sampling
1.2. Estimating the probability function (Law of Large Numbers)
2. Random Variables as Probabilistic Models: Counting measures (from sample data to theoretical properties)
2.1. Repeated and independent trials - Bernoulli scheme
2.2. First success trials
3. Introduction to Stochastic Processes
3.1. The Bernoulli and Binomial Processes
3.1.1. From trajectory samples to the frequency distribution of the first success trial (geometric and shifted geometric distributions)
3.1.2. Estimating interarrival times and distribution properties
3.2. The Poisson Process
3.2.1. Its characterizations
3.2.2. The time of the first success
4. Random Walk
4.1. From the binomial process to the simple random walk
4.2. Estimating the distribution of the first return time and the number of returns
4.4. Rescaling a simple random walk and defining limit processes
5. Markov Chains
5.1. Properties through sampling
6. Applications: From biomathematics to chemical reactions. Quantifying uncertainty
1.1. Sampling
1.2. Estimating the probability function (Law of Large Numbers)
2. Random Variables as Probabilistic Models: Counting measures (from sample data to theoretical properties)
2.1. Repeated and independent trials - Bernoulli scheme
2.2. First success trials
3. Introduction to Stochastic Processes
3.1. The Bernoulli and Binomial Processes
3.1.1. From trajectory samples to the frequency distribution of the first success trial (geometric and shifted geometric distributions)
3.1.2. Estimating interarrival times and distribution properties
3.2. The Poisson Process
3.2.1. Its characterizations
3.2.2. The time of the first success
4. Random Walk
4.1. From the binomial process to the simple random walk
4.2. Estimating the distribution of the first return time and the number of returns
4.4. Rescaling a simple random walk and defining limit processes
5. Markov Chains
5.1. Properties through sampling
6. Applications: From biomathematics to chemical reactions. Quantifying uncertainty
Prerequisites for admission
A basic course in Probability
Teaching methods
Computer lab sessions and interactive lectures.
Teaching Resources
V. Capasso, D. Morale, Una Guida allo studio della Probabilità e della Statistica Matematica, Esculapio editore
J.R. Norris - Markov Chain - Cambridge series on statistical and probabilistic mathematics, 1998
The course is highly interactive: students will prepare their own notes through the homework assignments
J.R. Norris - Markov Chain - Cambridge series on statistical and probabilistic mathematics, 1998
The course is highly interactive: students will prepare their own notes through the homework assignments
Assessment methods and Criteria
The exam consists of submitting a series of homework assignments that will be given by the instructors during the course. These assignments involve solving problems related to stochastic processes and their properties. Attending the course in real time is necessary to complete the homework, therefore attendance is strongly recommended.
Non-attending students will be required to take an oral exam covering the entire course program as well as specific homework assignments prepared for them.
The evaluation will assess the student's ability to analyze and draw conclusions from simulated samples and to deduce theoretical properties. The final grade will be Pass / Fail.
Non-attending students will be required to take an oral exam covering the entire course program as well as specific homework assignments prepared for them.
The evaluation will assess the student's ability to analyze and draw conclusions from simulated samples and to deduce theoretical properties. The final grade will be Pass / Fail.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 3
Laboratories: 36 hours
Professors:
Morale Daniela, Ugolini Stefania
Shifts:
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