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 T1).

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:
T1. Probability and statistics for computer scientists. Baron, Michael, 2. ed. : Chapman & Hall/CRC, 2014
T2. 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 a written final exam with a maximum value of 30 points. The exercises and questions are similar to those solved during the course or included in Moodle. A mock exam and some exams for the past academic year will be available for reference on the Moodle page for the course.
During the exam, students may use a formulary (both sides of a single A4 sheet) and the software R (an essential part of the program and subject to examination) for all necessary calculations. The use of a calculator or other devises will not be allowed.
The exam will have a total duration of 90 minutes, divided into 2 parts:
- (10-15 points) a moodle quiz (with automatic correction) composed of single choice queries and numerical exercises
- (15-20 points) theoretical questions and exercises with solutions to be submitted in RMarkdown format
Further details on Moodle.

Grading scale (for attending and non-attending students alike):
- Sufficient (18-22 points): to students who demonstrate a sufficient theoretical knowledge base and capacity to apply the concepts covered throughout the course, as well as a sufficient capacity to elaborate and present results using the specific language and mathematical notation associated to probability models and their interpretation
- Good (23-26 points): to students who demonstrate a good theoretical knowledge base and capacity to apply the concepts covered throughout the course, as well as a good capacity to elaborate and present results using the specific language and mathematical notation associated to probability models and their interpretation
- Very good (23-26 points): to students who demonstrate a very good superior theoretical knowledge base and capacity to apply the concepts covered throughout the course, as well as a very good or superior capacity to elaborate and present results using the specific language and mathematical notation associated to probability models and their interpretation and at least a basic capacity to identify relations between different concepts covered throughout the course and formulate independent judgement.
- Honors will be granted to students exhibiting an excellent knowledge base anc capacity to apply the concepts covered during the course through the use of specific language and mathematical notation, including the identification of relationships between different concepts and definitions.

Students may improve their final grade (iup to a maximum of 3 extra points, to be added to the final written exam mark) thorough substantial and continuative work during the course. Such work will be evaluated through the (non-mandatory) participation, exclusively in presence, to quizzes (on Moodle and about 10-25 minutes long) on pre-defined dates (to be published on Moodle during the first two weeks of lessons). Voluntary participation in a team project may give access to additional extra points (up to 6). 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.
Any extra points earned 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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