PROBABILITY AND DATA SCIENCE
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- GIACOMO DIMARCO
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/08
Training objectives
- The aim of the course is to provide the basic ideas of probability and statistics, and to show how they can be applied to an elementary study of various phenomena in different scientific areas such as ingeneering, economics, physics.
It is useful to recall that the theory of probability is preliminary to the study of random signals, argument which is treated in the course of signals and communications.
The main knowledge acquired regard:
Basic elements of descriptive statistics.
Basic elements of the theory of probability.
The main skills acquired regard:
understanding of topics in probability and statistics.
Ability to identify a probabilistic model and understand its main features.
Ability to inductive and deductive reasoning in addressing problems involving random phenomena.
Ability to outline a random phenomenon, setting a problem and solve it using appropriate tools of probability and statistics.
Making judgments and critical reasoning.
Ability to discuss about topics of probabilistic-statistical nature. Ability to acquire and manage new information about models in the presence of randomness. Prerequisites
- The knowledge of basic calculus such as real numbers, sequences and fundamental limits, main functions (polynomial, exponential, circular functions), differential calculus and integration theory.
Knowledge of the algebraic structure of real numbers. Knowledge of sequences, functions (polynomial, trigonometric, exponential and their inverses) and of their fundamental limits. Knowledge of differential and integral calculus.
Some of differential and integral calculus in several variables is also necessary. These concepts are taught during the course in parallel of Analysis Ib. Course programme
- Introduction to Statistics. Descriptive Statistics. Describing data sets. Summarizing data sets. Chebyshev’s Inequality. Normal data sets. Paired data sets and sample correlation coefficient. (5 hours)
Combinatorial analysis. The basic principle of counting. Permutations, combinations. Multinomial coefficients. The number of integer solutions of equations. (5 hours)
Elements of Probability. Sample space and events. Axioms of probability. Sample spaces having equally likely outcomes. Conditional probability and independence. Bayes formula. (10 hours)
Discrete Random Variables. Discrete density. Expected value. Expectation of a function of a random variable. Variance. Bernoulli and Binomial random variables. Poisson random variable. Geometric random variable. Negative binomial random variable. Hypergeometric random variable. Expected value of the sum of random variables. (10 hours)
Continuous Random Variables. Density, distribution function, expectation and variance of continuous random variables. Functions of continuous random variables. Uniform random variable. Normal random variable. Exponential random variable. Approximation of a binomial by a normal random variable. Distribution of a function of a random variable. (10 hours)
Jointly distributed Random Variables. Joint distribution functions. Independent random variables. Sum of independent random variables. Sum of normal random variables. Conditional distributions. Joint probability distribution of functions of random variables. (7.5 hours)
Properties of Expectation. Simple properties. Expectation of the sum of random variables. Moments of the number of events that occur. Covariance, variance of sum and correlation. Conditional expectation. Moment generating functions. (5 hours)
Limit Theorems. Markov inequality, Chabyshev inequality. Weak and strong law of large numbers. The central limit theorem. (5 hours)
Estimation Theory and Regression: Introduction. (2.5 hours) Didactic methods
- Theoretical and practical lessons. The exercises are carried out togheter with students. In particular, the instructor requires attending students to propose solutions to the exercises.
Learning assessment procedures
- The aim of the exam is to verify at which level the learning objectives previously described have been acquired. It consists of one written examination and of one oral examination.
Written examination
The written test consists in the resolution of some exercises, such as those done during classes or proposed as examples or as exercises in the textbook. The use of notes and consultation of texts is allowed. Students are admitted to the oral examination with a minimum score of 15. If the written test is repeated, the previous vote is canceled.
Oral examination.
The oral test is mandatory The oral examination typically changes the score of the written exam in a range of [-6, +6]. If the oral examination is not sufficient and the written test also, the full exam must be repeated. The oral examination should be done in the same exam session of the written test.
All detailed information can be found on the course classroom page. Reference texts
- S. M. Ross, Calcolo delle Probabilità, Apogeo 2014.
(Reference text for probability topics. Chapters 1-8).
S. M. Ross, Probabilità e statistica per l'ingegneria e le scienze, Apogeo 2003.
(Reference text for descriptive statistics and estimation theory. There are equally treated all probability topics discussed during the course but with less details. Chapters 1-7).
R. Spiegel, Probabilità e statistica: 760 problemi risolti, collana Schaum teoria e problemi, ETAS libri.
(Text for exercises, only if the student fells the need to do additional exercices with respect to the ones given by the instructor).
Exercices given by the instructor available on the web site
http://www.unife.it/ing/informazione/Calcolo-Statistica/materiale-didattico
