24
Mathematical Analysis
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
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.
Course Prerequisites
In-depth knowledge of the topics covered in Analysis I
Teaching Methods
Lectures and practical exercises (also with the help of MATLAB)
Assessment Methods
The exam, of about 4 hours, includes three questions from the Numerical Methods section and four from the Probability section. In each case, one of the questions requires the development of a theoretical argument.
The Numerical Methods module may include a computer-assisted numerical test.
The final score for each of the two sections (out of 30) is determined based on the overall quality of the proposed solutions. Specifically:
18–20: elementary and essential solution, without theoretical arguments;
21–24: substantially correct solution of most exercises, with limited or fragmentary theoretical arguments;
25–27: substantially correct solution of all exercises, accompanied by some good theoretical arguments;
28–30: complete and correct solution of all exercises, supported by clear and rigorous arguments.
The scores obtained in the two sections are combined and averaged. The average result provides the overall grade, expressed out of thirtieths.
Honors are awarded for a complete exam, correct in every part, free of inaccuracies, and demonstrating full mastery of theoretical tools and the ability to present clear and well-structured arguments.
Texts
S.M.Ross, Calcolo delle Probabilità, Terza edizione, Apogeo.
Contents
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.
More information
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