APPLIED PROBABILITY

This course is part of the required interdisciplinary activities of the Master's degree programme in Computer Science. Its aim is to provide the student with the fundamental tools of Probability at the basis of data analysis and mathematical modelling in the presence of uncertainty. The student will acquire quantitative skills and knowledge of some of the basic probabilistic models and software used to describe and analyze relevant processes in the field of Computer Science, among others.

Attendance and participation in the training activities of the course, together with individual study will allow students to:

1. Knowledge and understanding:
- know and understand the probability models that serve as a foundation for advanced methods of statistical learning and data analysis
- know and understand, in particular, Markovian probability models and the foundations of some stochastic processes used to represent dynamic phenomena in the presence of uncertainty

2. Ability to apply knowledge and understanding:
- use of specific programs for simulation and to calculate probabilities for the main families distributions
- capacity to autonomously analyze the properties of Markov chains, identifying their implications
- use of the appropriate formulas and terminology when applying and communicating the acquired knowledge

3. Ability to judge:
- contextualizing the acquired knowledge by identifying the most suitable models and methods for each situation

4. Communication skills:
- clear and exhaustive presentation of the results obtained as a solution to a probabilistic problem, using rigorous formulas and appropriate terminology

5. Learning skills:
- use and integrating information from notes, books, slides and practical sessions
- evaluation of the individual skills and preparation via quizzes and self-assessment exercises assigned during the course

Working differentiation and integration skills at the level of standard undergraduate calculus courses (a refresher, for reference purposes only is available in Section 12.3 of the textbook).

Basic matrix computations at the level of standard undergraduate linear algebra courses, in particular matrix multiplication and inversion and, solving linear systems of equations (a refresher, for reference purposes only is available in Section 12.4 of the T1 textbook).

Basic knowledge of probability at the level of a Bachelor in Computer Science is advised. In particular, events, axioms of probability, conditional probability and independence, random variables, expected value, variance, covariance and correlation, main discrete and continuous distributions, central limit theorem, law of large numbers (these subjects, covered in chapters 2-3 of the T1 textbook will be reviewed during the course)

1. Reminder of the basic concepts of Probability and Random Variables
- Axiomatic probability, conditional probability and independence
- Discrete Random Variables and Their Distributions
- Continuous Distributions
- Random vectors: joint, marginal and conditional distributions
2. Stochastic Processes
- Markov processes and Markov chains
- Discrete time Markov chains
- Counting processes
- Continuous time Markov chains
- Poisson process
- Simulation of stochastic processes

Main textbooks:
1. Probability and statistics for computer scientists. Baron, Michael, 2. ed. : Chapman & Hall/CRC, 2014
2. Probability with Applications in Engineering, Science, and Technology. Carlton, Matthew A. and Devore, Jay L., 2 ed.: Springer, Cham, 2017

Additional resources:
Additional suggested reading and materials made available on the Moodle platform

Achievement of the course objectives is evaluated through participation in activities and assignments during the course together with a team project and a written final exam.

The written final exam has a value of 30 points. The exercises are similar to those found in the textbooks or those solved during the course or included in Moodle.
During the exam, the use of a formulary and computer is allowed.
The use of the software R is an essential part of the program and is subject to examination.

The team project has a maximum value of 6 points. The student's capacity to form and work in teams will be part of the evaluation: each student is responsible for finding their own team and all team members will receive the same mark. Individual assignments will NOT be allowed. Further details on the format and submission deadlines for the project will be available on Moodle.

Frequenting students may accumulate up to 3 extra points, to be added to the final exam mark, by participating in class exercises and activities and via Moodle quizzes taking place every pre-specified dates (available on Moodle).

Any extra points earned and points for team assignment remain valid for all 4 exams of the academic year, but are lost and no longer valid for future exams if the student renounces a passing grade.

Theoretical lectures and exercises, including practical sessions using the software R. Use of Moodle platform for learning assessment.

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