- Docente: Roberta Nibbi
- Credits: 6
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: First cycle degree programme (L) in Engineering Management (cod. 0925)
Learning outcomes
A sound theoretical basis as well as a working knowledge of the fundamental mathematical methods aimed at coping with uncertainty in physical and other phenomena.
Course contents
Foundations of probability theory.Events and sets.
Kolmogorov's axioms. Joint probability, conditional probability,
independence.
Total probability theorem and Bayes' theorem.
Random variables.Discrete and continuous random
variables. Cumulative distribution function. Continuous random
variables with probability density. Characteristic numerical values
of random variables: expected value (mean), variance, standard
deviation, mean square error, moments. Pairs and vectors of random
variables: joint and marginal cumulative distribution functions,
joint and marginal probability densities. Laws of conditional
distribution, independence. Characteristic numerical values: mean
values, covariance matrix, moments. Correlated and uncorrelated
random variables.
Models of random variables. Bernoulli scheme. Binomial, Poisson,
uniform, normal, exponential random variables. Relationships among
some of these kinds of random variables.
Functions of random variables.Characteristic
numerical values: representation of the expected value and of the
variance, with applications to some notablecases (sum and product
of two random variables, linear combination of a finite number of
random variables, case of independent, identically distributed
random variables, etc.). Notions on the determination of the
probability distribution for a function of one or more random
variables.
Limit theorems in probability.Sequences of random
variables and notions of convergence. Markov inequality, Chebyshev
inequality. Laws of large numbers. Central limit theorem.
Introduction to statistics.Sample mean, median and
mode, sample variance and standard deviation, percentiles.
Bivariate data sets and sample correlation coefficient. Statistical
inference. Sampling. Estimators and confidence intervals,
efficiency of point estimators. Hypothesis testing. Linear
regression.
Readings/Bibliography
H. Hsu, Probabilità, variabili casuali e processi stocastici, ed.
McGraw-Hill Italia.
P. Erto, Probabilità e statistica per le scienze e l'ingegneria
3/ed, ed. McGraw-Hill Italia.
A. M. Mood, F. A. Graybill, D. C. Boes, Introduzione alla
statistica, ed. McGraw-Hill Italia.
M. Giorgetti, E. Mazzola, Probabilità e Statistica matematica, ed.
Pearson (eserciziario).
Teaching methods
Standard lectures held by the teacher alternating with exercise classes.
Assessment methods
A midterm and a final written tests, containing also theoretical
questions, will be offered during the course.
The course is passed if the mean of the marks obtained in the
midterm and final exams is greater or equal to 18/30, in which case
the finale grade will be the arithmetic mean of these two grades.
In addition, an oral exam may be taken if requested by the student,
but only after passing the written exam.
A comprehensive written exam, containing also theoretical
questions, may be taken after the end of the course by those
students who did not take or pass the midterm and/or the final
test.
Again, an additional oral exam is available on the student's
request after passing the written part.
Teaching tools
Blackboard, slides and projector.
Office hours
See the website of Roberta Nibbi