Insegnamento a.a. 2024-2025

• Conditional probabilities and conditional expectations.
• Introduction to stochastic processes and Markov chains.
• Discrete-time Markov chains: Chapman-Kolmogorov equation, Classification of states, Limiting properties, Applications (e.g. stochastic models, sequential testing, website ranking).
• Introduction to Stochastic Simulation, Simulation techniques and Monte Carlo methods.
• Markov Chain Monte Carlo algorithms, Computational applications.
• Counting processes and the Poisson process, Continuous-time stochastic processes, Examples and modeling applications.

• Understand the connection between simple random variables and more elaborate and realistic stochastic processes.
• Formulate and analyse probabilistic models based on Markov chains and counting processes.
• Understand the principles of stochastic simulation and Monte Carlo algorithms.

• Translate real-world situations involving randomness and uncertainty into probabilistic models.
• Exploit the tools of probability theory to answer relevant questions, such as: what is the equilibrium distribution of a specific stochastic system? What is the probability of a certain outcome?
• Use software to simulate the evolution of probabilistic systems. Design simple Monte Carlo algorithms for computational purposes.
• Apply stochastic processes to model and solve real-world problems commonly occurring in diverse fields, such as business, industry, economics, finance and data science.

• Practical Exercises
• Individual works / Assignments

The main learning activity of this course consists in face-to-face lectures. Lectures integrate discussions of theoretical notions, such as mathematical definitions, with examples and in-class solution of exercises. The aim of the examples and exercises shown in class is two folded:

• First to illustrate and visualize the theoretical notions with simple examples.
• Secondly to describe real-world applications (both modeling and computational) of the probabilistic tools.

In addition, the R statistical software is used in class to display computer simulations of the stochastic models under consideration. A set of exercises are provided for students to practice and get familiar with the lectures’ content.  

The core assessment for all students, both attending and non-attending, consists in a written exam. The aim of the exam is to verify both the students’ understanding of the theoretical material (e.g. ability to provide definitions and characterizations of some noteworthy stochastic process), and their capacity to apply the theory for modeling or computational purposes (e.g. ability to identify appropriate models for a given situation and use them to answer specific questions).

Additionally, throughout the course 4 individual assignments will be made available. Assignments submitted before the deadline are marked and can grant up to 3 bonus points in the final score for the module.

S.M. ROSS, Introduction to probability models, Academic Press, 2014.