Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate

SUBJECT

Code
Subject
MMP0001 Probability Theory
Section
Semester
Hours: C+S+L
Category
Type
Mathematics
4
2+2+0
speciality
compulsory
Mathematics and Computer Science
4
2+2+0
speciality
compulsory
Teaching Staff in Charge
Assoc.Prof. LISEI Hannelore-Inge, Ph.D.,  hannemath.ubbcluj.ro
Assoc.Prof. SOOS Anna, Ph.D.,  asoosmath.ubbcluj.ro
Asist. ROSCA Natalia Carmen, Ph.D.,  nataliamath.ubbcluj.ro
Aims
The use of basic facts of the probability theory in some applications.
Content
1.Probability space. Experiments and events. Sigma fields and probabilities. Elementary basic formulas. Geometric probability. Conditional probability.Independence of events.
2. Classic probabilistic models.Bernoulli models: samplings with replacement, samplings without replacement. Poisson model. Pascal model.
3. Random variables. Definition and properties. Distribution function. Density function. Discrete and continuous distributions. Joint distribution and density function. Marginal distributions and marginal densities. Independent random variables, properties.
4. Numerical characteristic of random variables. Expectation, properties. Moment of the k-th order , absolute moment of the k-th order , central moment of the k-th order. Variance. Correlation and correlation coefficient, properties. Inequalities.
5. Sequences of random variables. Convergence in probability, in distribution, almost surely, in mean of order r. Properties.
6.Characteristic functions.Definition and properties. Characteristic functions of some classical distributions. Inversion formula. Positive semi-definite functions. Bochner-Khinchin theorem.
7. Laws of large numbers. Weak law of large numbers. Theorems: Markov, Chebyshev, Poisson, Bernoulli. Strong law of large numbers. Kolmogorov theorem.
8. Limit theorems. Lindeberg condition and the central limit theorem. Lyapunov theorem. Moivre Laplace theorems. Applications.
References
1. AGRATINI, OCTAVIAN: Capitole speciale de matematici, Lito., Universitatea $Babeş-Bolyai$ Cluj-Napoca, 1996.
2. BLAGA, PETRU: Calculul probabilităţilor şi statistică matematică, Vol. II, Curs şi culegere de probleme, Universitatea $Babeş-Bolyai$ Cluj-Napoca, 1994.
3. CIUCU, G., CRAIU, V., SĂCUIU, I.: Probleme de teoria probabilităţilor. Bucureşti: Editura Tehnică, 1974.
4. DUMITRESCU, M., FLOREA, D., TUDOR, C.: Probleme de teoria probabilitătilor şi statistică matematică. Bucureşti: Editura Tehnică, 1985.
5. FELLER, W.: An introduction to probability theory and its applications, Vol.I-II. New York: John Wiley, 1970-1971.
6. GNEDENKO, B.V.: The theory of probability. Moscow: Mir Publishers, 1976.
7. IOSIFESCU, M., MIHOC, GH., THEODORESCU, R.: Teoria probabilităţilor şi statistică matematică. Bucuresti: Editura Tehnică, 1966.
8. LISEI, HANNELORE: Probability Theory. Cluj-Napoca: Casa Cărţii de Ştiinţă, 2004.
9. MIHOC, ION: Calculul probabilităţilor şi statistică matematică. P. I-II: sCluj-Napoca: Universitatea $Babeş-Bolyai$ Cluj-Napoca, 1994.
10. SHIRYAEV, A.N.: Probability. New York: Springer (2nd ed.), 1995.
Assessment
During the semester: a Control Paper.
In the session: written exam.
The final grade: the arithmetic mean of the above 2 marks having the weights 1 and 3, respectively.
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject