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	
Applied Mathematics	4	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Prof. AGRATINI Octavian, Ph.D., agratini@math.ubbcluj.ro
Assoc.Prof. SOOS Anna, Ph.D., asoos@math.ubbcluj.ro
Assoc.Prof. LISEI Hannelore-Inge, Ph.D., hanne@math.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.  	
Bibliography	 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.