## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Probability Theory
 Code Semes-ter Hours: C+S+L Type Section MC003 6 2+2+0 compulsory Matematica MC003 6 2+2+0 compulsory Matematică-Informatică MC003 6 2+2+0 compulsory Matematici aplicate
 Teaching Staff in Charge
 Prof. AGRATINI Octavian, Ph.D.,  agratinimath.ubbcluj.roAssoc.Prof. SOOS Anna, Ph.D.,  asoosmath.ubbcluj.roLect. LISEI Hannelore-Inge, Ph.D.,  hannemath.ubbcluj.ro
 Aims To acquire basic knowledge of Probability Theory, focusing on applications.
 Content 1. Field of events. Random events. Sample space. Relationships between events. Operations with events, properties. Classification of events. Partitions. 2. Probability space. Classical definition of probability, axiomatic definition of probability. Finite probability space. Properties of a finite probability space. Infinite (Borel) probability space. Properties of an infinite probability space. Conditional probability. Properties. Total probability formula, Bayes formula. Independent events. Properties. Boole@s inequality. The Borel-Cantelli lemma. 3. Classical probabilistic models. Bernoulli model with replacement with two or more states. Bernoulli model without replacement with two or more states. Poisson model. Pascal model. Markov-Polya model. 4. Random variables and random vectors. Discrete random variables. Discrete probability laws (binomial, hypergeometric, Poisson, Pascal, geometric, multinomial). Independence. Operations with discrete random variables. Distribution function. Properties. Continuous random variables. Probability density function. Properties. Continuous probability laws(uniform, normal, Gamma, exponential, Chi-squared, Student, Cauchy, Beta, Fisher-Snedecor). Independence. Operations with continuous random variables. 5. Numerical characteristics of random variables. Expectation, properties. Variance, properties. Moments (initial, central, absolute, factorial). Properties. Covariance, (correlation), correlation coefficient. Properties. Other characteristics: median, mode, quantile, skewness, kurtosis. Inequalities: Hölder, Schwarz, Cauchy-Buniakovski, Minkovski, Chebyshev. 6. Sequences of random variables. Convergence in probability. Strongly convergence. Almost surely convergence. Convergence in distribution. Comparison between different types of convergence. 7. Characteristic functions. Properties. Inversion formula. Uniqueness theorem. Convergence theorem for characteristic functions. Positive semi-definite functions. Bochner-Hincin theorem. 8. Laws of large numbers. The weak law of large numbers: Markov@s theorem, Chebyshev@s theorem, Poisson@s theorem, Bernoulli@s theorem, Hincin@s theorem, Kolmogorov@s theorem. 9. Limit theorems. Lindeberg@s theorem, Lyapunov@s theorem. The local and integral Moivre-Laplace theorems. Other limit theorems.
 References 1. BLAGA, PETRU: Calculul probabilitatilor. Culegere de probleme. Cluj-Napoca: Universitatea \$Babeo-Bolyai\$ Cluj-Napoca, 1984. 2. BLAGA, PETRU - RADULESCU, MARCEL: Calculul probabilitatilor. Cluj-Napoca: Universitatea \$Babeo-Bolyai\$ Cluj-Napoca, 1987. 3. CIUCU, G. - CRAIU, V. - SACUIU, I.: Probleme de teoria probabilitatilor. Bucuresti: Editura Tehnica, 1974. 4. DUMITRESCU, M. - FLOREA, D.- TUDOR, C.: Probleme de teoria probabilitatilor oi statistica matematica. Bucureoti: Editura Tehnica, 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 probabilita?ilor oi statistica matematica. Bucuresti: Editura Tehnica, 1966. 8. LISEI, HANNELORE: Probability theory. Cluj-Napoca: Casa Cartii de atiinta, 2004. 9. MIHOC, ION: Calculul probabilitatilor oi statistica matematica. P. I-II: sCluj-Napoca: Universitatea \$Babes-Bolyai\$ Cluj-Napoca, 1994. 10. SHIRYAEV, A.N.: Probability. New York: Springer (2nd ed.), 1995.
 Assessment The final grade will be compiled of the following: - final exam at the end of the semester, consisting of: - the written exam; the points obtained for activity in seminar will be added to the score obtained at the written exam. - the oral exam, for students who did not pass the written exam, or for those wishing to improve their written exam grade, in which case the grade will be the average of the two (written and oral) scores; the points obtained for activity in seminar will be added to this score.
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject