PROBABILITY AND RANDOM VARIABLES
Learning outcomes of the course unit
The course aims at providing the student
with a basic knowledge of probability
theory and random variables, with
applications to Engineering.
Geometria, Analisi A
Course contents summary
Probability theory: concepts from set theory,
axioms of probability theory and their consequences.
Elements of combinatorics. Conditional
probability, total probability theorem
and Bayes formula. Repeated trials.
Random variables: introduction to the
concept of probability density function.
Formal definition of probability density
function and the cumulative
distribution function. Dirac delta.
Continuous and discrete random variables.
Trasformations of random variables:
trasformation of a single random variable and
fundamental theorem. Expected value and Law of
the Unconscious Statistician (LUS).
Moments and moment generating function.
Mixed Bayes formula and continuous version of the
total probability theorem. Pairs of random
variables and their transformations. Extensions to systems of n random
variables. Generalization of the LUS and conditional
expectation theorem for n random variables.
Correlation. Independence and incorrelation.
Law of large numbers and its statistical
interpretation. Statistical interpretation
of the covariance. Correlation coefficient. Central
limit theorem. De Moivre-Laplace theorem.
More information at:
Weekly assignment of homework problems to
the students, without formal grading.
Solutions available in the textbook.
A. Bononi e G. Ferrari: "Teoria della probabilità e variabili aleatorie
con applicazioni", McGraw-Hill-Italia, marzo 2005,
Extra solved problems
G. Prati: "Esercizi di teoria delle variabili casuali"
(collection of solved exercises).
A. Papoulis: "Probability, Random variables, and stochastic
processes", McGraw-Hill, 3rd Ed., 1991.
The exam is written only. Duration: 3 hours. The grade of
the written exam, if not smaller than 18, is
registered as the final grade of the exam,
except for special cases, at
the teacher's discretion, in which an additional
oral exam may be requested. The grade of
the written exam must be registered
before the next exam date, and expires after that date.
During the written exam,
one is allowed to bring:
1) a calculator;
2) an A4 sheet of paper with formulas.
During the semester, a midterm exam (around the end of April)
and an endterm exam (around the middle of June) will be held. All students
(regardless of the immatriculation year) can partecipate.