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

 Theory of probability
 Code Semes-ter Hours: C+S+L Credits Type Section MC014 6 2+2+0 6 compulsory Matematica Economica MC014 6 2+2+0 6 compulsory Matematici Aplicate
 Teaching Staff in Charge
 Prof. BLAGA Petru, Ph.D., pblaga@cs.ubbcluj.roLect. SOOS Anna, Ph.D., asoos@math.ubbcluj.ro
 Aims Knowledge of basic concepts of probability theory required to approach the study of stochastic processes and mathematical statistics.
 Content 1. Sample space and events. Probability space: Relations among events, operations with events. Definition of probability. Kolmogorov's axiomatization. Conditional probability, total probability and Bayes formulas. Independence. Classical schemes of probability (Bernoulli, Poisson, Pascal, geometric, Markov-Polya). 2. Random variables and random vectors and their properties: Discrete random variable. Discrete probability laws (Bernoulli, binomial, hypergeometric, Poisson, negative binomial, geometric, Markov-Polya, multinomial, multidimensional hypergeometric, multidimensional Poisson). Distribution function. Continuous random variable. Continous probability laws (uniform, normal, lognormal, gamma, exponential, chi square, Student, Fisher-Snedecor,beta). Marginal distribution. Independence of variables. Conditional distribution. Function of random variable. Function of random variables. 3. Numerical characteristics of variables: Expectation. Variance. Moments (initial, central, and factorial). Covariance. Coefficient of correlation. Median. Quantiles. Skewness. Kurtosis. Conditional expectation and variance. Inequalities (Holder, Schwarz, Cauchy-Buniakovski, Minkovski, Chebyshev, Kolmogorov). 4. Characteristic function: Inversion formula. Positive semi-definite functions, Bochner-Hincin theorem. Convergence theorem for sequence of characteristic functions. 5. Sequences of random variables: Weak convergence, strong convergence, almost sure convergence, convergence in law. 6. Law of large numbers: Weak law of large numbers. Strong law of large numbers. Bernoulli, Poisson, Hincin, Chebyshev, Markov, and Kolmogorov theorems. 7. Limit theorems: Asymptotic problem. Central asymptotic problem. Central limit theorems (Lindeberg, Lyapunov, Moivre-Laplace)
 References 1. Blaga, P., Calculul probabilitatilor. Culegere de probleme, lito. Univ. "Babes-Bolyai" din Cluj-Napoca, 1984. 2. Blaga, P., Muresan A. S., Matematici aplicate in economie, Vol.I, Transilvania Press, Cluj-Napoca, 1996. 3. Blaga, P., Radulescu, M., Calculul probabilitatilor, lito. Univ. "Babes-Bolyai" din Cluj-Napoca, 1987. 4. Ciucu, G., Craiu, V., Introducere in teoria probabilitatilor si statistica matematica, Ed. did. si ped., Bucuresti, 1971. 5. Ciucu, G., Tudor, C., Teoria probabilitatilor si aplicatii, Ed. st. si encicl., Bucuresti, 1983. 6. Gnedenko, B. V., The theory of probability, Mir Publishers, Moscow, 1976. 7. Iosifescu, M., Mihoc, Gh., Theodorescu, R., Teoria probabilitatilor si statistica matematica, Ed. tehnica, Bucuresti, 1966. 8. Mihoc, I., Calculul probabilitatilor si statistica matematica, Partea I si II, lito. Univ. "Babes-Bolyai" din Cluj-Napoca, 1994,1995. 9. Rao, M. M., Probability Theory with Applications, Academic Press, New York, 1984. 10.Shiryaev, A. N., Probability (Second Edition), Springer, Heidelberg, 1995.
 Assessment Exam.