Mathematical Sciences for Artificial Intelligence

Sample spaces, set operations, axioms of probability, equiprobable sample spaces. Exercises. 8 hours. Combinatorial analysis: fundamental principle of counting, arrangements, simple arrangements, permutations, binomial coefficient, multinomial coefficients, sampling from urns with and without replacement, urns with one or two types of balls, problems with random permutations (“hat problems”). Exercises. 8 hours. Conditional probability, product rule, total probability, Bayes’ theorem. Exercises. 8 hours. Independence of events, Bernoulli trials, sequences of increasing and decreasing events, probability of all successes, probability that the first trial is a success. Exercises. 8 hours. Random variables, distribution function, discrete random variables, discrete density, relation between discrete density and distribution function, expectation, notable random variables: Binomial, Poisson, Poisson as a limit of Binomials, Hypergeometric. Exercises. 8 hours. Sum of expected values. Continuous random variables. Expected value and variance. Notable continuous random variables: uniform, Gaussian, exponential. Exercises. 8 hours. Joint laws of random variables, expectation of functions of several random variables, independent random variables, expectation of the product of independent random variables, covariance and its properties. Sum of independent random variables: notable cases and convolution. Conditional density and distribution: definition and example. Conditional expectation and computation of expectations via conditioning. Conditional variance and computation of variance via conditional variance. Exercises. 12 hours. Markov and Chebyshev inequalities and examples. Law of large numbers, central limit theorem. Exercises. 7 hours. Descriptive statistics: histograms, measures of central tendency (sample mean, mode, sample median), sample variance, bivariate data, correlation coefficient, difference between correlation and causation. Inferential statistics: maximum likelihood estimation of the mean for i.i.d. Bernoulli, Gaussian, and Poisson data; confidence intervals for the mean in the case of i.i.d. Gaussian data with known and unknown variance. Exercises. 11 hours. Review exercises and self-assessment tests: 7 hours.

Students are expected to have a basic background in mathematics, typically acquired in first-semester courses or during secondary education. In particular: -fundamental knowledge of arithmetic and algebra (manipulation of expressions, equations, inequalities); -familiarity with basic analytic geometry (lines, planes, Cartesian coordinates); -understanding of real functions of one real variable, especially elementary functions, their graphs, and main properties; -basic notions of differential calculus (limits and derivatives); -understanding of basic logical and set-theoretic concepts. No prior knowledge of probability or statistics is required.

Ross, Sheldon M. Calcolo delle probabilità. Edizione italiana a cura di Carlo Mariconda e Marco Ferrante. Milano: Apogeo, 2005. Collana: Idee & Strumenti. ISBN 978-88-503-2257-3. Traduzione italiana di A First Course in Probability, 7ª edizione, Pearson Prentice Hall.

Attendance is strongly recommended.

Written and oral exam.

Ross, Sheldon M. Calcolo delle probabilità. Edizione italiana a cura di Carlo Mariconda e Marco Ferrante. Milano: Apogeo, 2005. Collana: Idee & Strumenti. ISBN 978-88-503-2257-3. Traduzione italiana di A First Course in Probability, 7ª edizione, Pearson Prentice Hall.

The course includes frontal lectures in which the instructor explains the fundamental concepts of probability and statistics, exploring both methodological and applied aspects. In addition to theoretical lectures, there are exercise sessions aimed at consolidating the understanding of the topics covered and developing the ability to apply theoretical notions to problem solving. Some exercises will be carried out in class together with the instructor, while others will be assigned as homework for individual practice.