PROBABILITY AND STATISTICS
*) Knowledge and understanding: the student must demonstrate knowledge and ability to understand the fundamental principles of the laws of probability and statistics.
*) Ability to apply knowledge and understanding: the student must demonstrate to be able to solve problems of low and medium complexity of probability and statistics.
*) Autonomy of judgment: the student must be able to develop the ability to critically analyze the different problems of probability and statistics calculation with particular reference to different case studies from those proposed during the course.
*) Communication skills: The student must have the ability to express clearly and simply technical concepts and to use correctly the scientific language.
*) Learning skills: The student must be able to integrate and update knowledge using different sources.
It is necessary to have acquired and assimilated the following knowledge provided
from the courses of Mathematics I, MathematicsII
- elementary concepts of study of the functions of one and two variables;
- knowledge of the fundamental concepts related to matrices and vectors.
- knowledge of elementary integral calculus.
Axiomatic approach to probability. Events. Law of Probability. Probability spaces. Examples of probability spaces (discrete, continuous). Mutual exclusivity. Conditional probability. Conditional probability and independence. Bayes theorem. Chain rule. Total probability theorem. Independence between events. Combined experiments. Random variables. Cumulative distribution function (CDF). Continuous, discrete and mixed random variables. Probability density function (pdf). Mass density function (pmf). Examples of random variables (Bernoulli, binomial, geometric, Poisson, uniform, Gaussian, exponential, Laplace, Rayleigh, "mixture". Repeated experiments. The theorems of de Moivre-Laplace. Transformations of a random variable. Calculation of the CDF. Calculation of the pdf: fundamental theorem on the transformations of random variables, calculation of pmf, inverse problem, generation of a random variable with assigned CDF, generation of random numbers, synthetic characterization of a random variable, mean of a random variable, fundamental theorem, Variance and mean quadratic value Moments of a random variable Notable inequalities.
Pairs of random variables. CDF, pdf and joint pmf. Joint and marginal statistics. Pair of jointly Gaussian random variables. Independence for pairs of random variables. Synthetic characterization of a pair of random variables. Correlation measures. Vector space of random variables. Orthogonality. Correlation coefficient. Incorrelation. Vectors of random variables. Statistical characterization of n random variables (CDF, pdf, pmf). Transformations of n random variables. Independent random variables. Mean and moments of n random variables. Fundamental theorem of the mean. Correlation and covariance matrix. Incorrelation. Vectors of jointly Gaussian random variables. Law of Large Numbers. Central Limit Theorem. The Meaning of Statistics. Sample statistics. The statistical decision. Criterion of the Maximum Likelihood. Bayes criterion. Statistical hypotheses. Medium value test. Test on variance. Kolmogorov-Smirnov test. Chi-square test. Notes on Statistical Estimation. Descriptive Statistics. Gamma, Chi-Quadro, t Student t r.v. Sample Statistics, Parametric Estimate, Confidence Intervals. Hypothesis Test and Theories of the Statistical Decision.
PROBABILITY LAWS (17 h):
Axiomatic approach. Events. Law of Probability. Probability spaces. Examples of probability spaces (discrete, continuous). Mutual exclusivity. Conditional probability. Statistical independence. Bayes theorem. Chain rule. Total probability theorem.
RANDOM VARIABLES (22 h):
Random variables. Cumulative distribution function (CDF). Continuous, discrete and mixed r.v. Probability density function (pdf). Mass density function (pmf). Examples of r.v. (Bernoulli, binomial, Poisson, uniform, Gaussian, exponential, Rayleigh, ... Repeated experiments. The theorems of de Moivre-Laplace. Transformations of a r.v.: CDF, pdf, and pmf. Synthetic characterization of a r.v., mean of a r.v., fundamental theorem, Variance and mean quadratic value Moments of a r.v.
VECTOR OF R.V. (18 h):
Pairs and vectors of r.v. Joint and marginal CDF, pdf and pmf. Independence of vectors of r.v. Joint moments of r.v. Correlation coefficient. Incorrelation. Orthogonality. Correlation and covariance matrices. Synthetic characterization of vectors of r.v. Vectors of jointly Gaussian r.v. Law of Large Numbers. Central Limit Theorem.
STATISTICS (15 h):
Descriptive Statistics, Gamma, Chi-Quadro, t Student r.v., Sample Statistics, Confidence Intervals, Hypothesis Test, Statistical Decision and Estimation.
The course is divided into frontal lessons and classroom exercises.
1) Sheldon Ross, "Probabilità e Statistica per l'Ingegneria e le Scienze", APOGEO Editore
2) Online Material
The objective of the examination is to verify the level of achievement of the previously indicated educational objectives. The exam is divided into 2 parts, a written test and an oral test, which take place a few days later.
- the written test consists in carrying out three probability and statistical exercises (random variables, distribution functions, repeated experiments, calculation of moments, transformation of random variables, pairs of random variables, probability laws, total probability theorem), with the objective to assess whether the student has the ability to analyze and solve them with the methodologies addressed during the course; The result of the written test will fall into the following grades: a (27-30); b (24-26); c (21-23); d (18-20); e (15-17); f (insufficient). The students can be take the oral test only if the written test result is at least "e". The time allowed for the written test is 2 hours. It is not possible to consult teaching material.
- an oral test in which the ability to connect and compare different aspects treated during the course will be evaluated;
The final grade is given by the weighted average of the 2 scores assigning 1/3 weight to the written test and 2/3 to the oral test.