Aim of the module is to provide a solid foundation on the fundamental principles of probability and the main discrete and continuous distributions. Students will acquire skills useful for solving concrete problems and develop the necessary knowledge to tackle subsequent applied courses.
In-depth knowledge of the topics covered in Analysis I
Lectures and practical exercises (also with the help of MATLAB).
The exam consists of a single written test lasting approximately 4 hours, aimed at verifying the acquisition of the required skills and the ability to apply them to the topics covered in the courses of Mathematical Analysis 2 and Probability Theory.
The test mainly involves solving exercises; at the instructor’s discretion, theoretical questions may also be included.
To pass the exam with the minimum grade of 18/30, the student must demonstrate at least an elementary level of knowledge and skills.
A grade between 20/30 and 24/30 is awarded when the student is able to solve part of the exercises almost correctly, while still showing basic competence.
A grade between 25/30 and 30/30 (with possible honors) is awarded when the student is able to solve all exercises correctly and demonstrates good argumentative skills.
S.M.Ross, Calcolo delle Probabilità, Terza edizione, Apogeo.
Foundations of Probability Calculus
Random phenomena. Sample space, events, sigma-algebras. Axioms of probability. Construction of probability measures: the classical definition. Elements of combinatorial calculus.
Conditional probability. Law of compound probability. Independent events. Law of total probability. Bayes’ theorem.
Random Variables
Definition of a random variable. Distribution function and its properties. Discrete random variables: probability mass function. Continuous random variables: probability density function. Expected value. Variance. Moments. Markov’s inequality. Chebyshev’s inequality. The three-sigma rule.
Random Variable Models
Discrete uniform distribution. Geometric distribution. Continuous uniform distribution. Exponential distribution. Binomial distribution. Poisson distribution. Poisson distribution as an approximation of the binomial distribution. Normal distribution.
Random Variables in R^n
Multidimensional random variables. Joint distribution function. Marginal distribution function. Discrete and continuous bivariate random variables. Independent random variables. Covariance. Uncorrelated random variables. Law of large numbers. Central limit theorem. Normal approximation of the binomial distribution.
EXPECTED LEARNING OUTCOMES
With reference to the Dublin Descriptors, the expected learning outcomes are as follows:
Knowledge and understanding The student knows the axiomatic foundations of probability theory, the concepts of conditional probability, independence, Bayes' theorem, the theory of discrete and continuous random variables (cumulative distribution function, density, expected value, variance, moments), and the main distribution models (uniform, exponential, binomial, Poisson, normal). Furthermore, the student is familiar with multidimensional random variables, covariance, and limit theorems.
Applying knowledge and understanding The student is able to calculate the probability of events using axioms, the law of total probability, and Bayes' theorem, determine the expected value and variance of random variables, recognize and utilize the main distributional models, and apply the law of large numbers and the central limit theorem to practical problems.
Making judgments The student can identify the most appropriate probabilistic model to describe a random phenomenon, choose the suitable calculation tools, and correctly interpret the results obtained in probabilistic terms.
Communication skills The student is able to present the solution to a probabilistic problem in a rigorous form, correctly using the formalism of events, random variables, and distributions.
Learning skills The student is capable of autonomously consulting probability and statistics textbooks and transferring the acquired tools to applied disciplines.
All teaching materials (handouts, lecture notes, and supplementary resources) will be made available on the dedicated Microsoft Teams channel; the access code will be provided at the beginning of the course.
