Study-unit PROBABILITY THEORY

Course name	Computer science and electronic engineering
Study-unit Code	70A00056
Curriculum	Comune a tutti i curricula
Lecturer	Paolo Carbone
Lecturers	Paolo Carbone
Hours	54 ore - Paolo Carbone
CFU	6
Course Regulation	Coorte 2023
Supplied	2024/25
Supplied other course regulation	Coorte 2022 - Will supplied in year 2023
Learning activities	Caratterizzante
Area	Ingegneria elettronica
Sector	ING-INF/07
Type of study-unit	Obbligatorio (Required)
Type of learning activities	Attività formativa monodisciplinare
Language of instruction	Italian
Contents	Introduction to probability theory: continuous, discrete and mixed random variables. Probability density functions. Probability distribution functions. Joint random variables. Introduction to stochastic processes: definitions, autocorrelation and autocovariance functions. Fundamentals of stochastic processes.
Reference texts	Roy. D. Yates, David J. Goodman, Probability and Stochastic Processes, John Wiley & Sons Inc; 2nd International Edition, 2004. Handouts by the instructor.
Educational objectives	The objective of this module is to provide the students with the knowledge to correctly apply probability theory and measurement theory. At the end of this class the student will have learned: - the concept of continuous and discrete random variable and its PDF and CDF - the concept of function of a random variable - the concept of joint random variables and vector of random variables - the concept of function of two random variables - the concept of stochastic process and of its general properties Moreover, he will be able to: - solve exercises that require modeling using discrete, continuous and mixed random variables - solve exercises using the concept of stochastic process
Prerequisites	Calculus I is mandatory. Students must also be able to perform simple mathematical modeling in two dimensions and capable to solve simple double integrals.
Teaching methods	Frontal lectures at the blackboard. Students are expected to provide active participation and show autonomous study capabilities. Proficiency in solving exercises can only be developed by complementing attendance of lectures with dedicated sessions in the solution of exercises at home.
Other information	Information about available services for people with disabilities and/or with learning disabilities, see: http://www.unipg.it/disabilita-e-dsa
Learning verification modality	Written and oral tests. The written examination consists in the resolution of exercises and multiple choice tests. The oral examination covers all course content program and consists both in questions about the theory and in the solution of exercises. Its length is about 20-25 minutes.
Extended program	Set theory. Sample spaces and random events. How to assign probabilities: classical, empirical and subjective approach. Conditional probability. Total probability theorem. Bayes theorem. Combinatorial calculus: permutations, dispositions, combinations. Random variables. Cumulative3 distribution function. Probability density function. Discrete random variables: Bernoulli, geometrical, binomial, Pascal, uniform discrete. Mode, median, expected value. Transformed random variables. Expected value of a transformed random variable. Variance and standard deviation. Central and non-central moments. Conditional mass probability. Continuous random variables. Cumulative and density functions. Expected value. Probability models: uniform, exponential, Gaussian. Mixed random variables. Transformed continuous random variables. Conditional continuous random variables. Couples of random variables. Marginal probability density functions. Transformation of two random variables. Rayleigh and Rice probability models. Orthogonal random variables. Correlation, covariance. Correlation coefficient. Conditioning two random variables. Random vectors. Gaussian random vectors. Central limit theorem. De-Moivre Laplace formula. Introduction to stochastic processes. Moments. Wide-sense and strict-sense stationary random processes. Ergodic processes.