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

Code	Subject	
MMP0003	Probability Theory and Statistics	

Section	Sem.	Hours: C+S+L	Category	Type	
Computer Science	3	2+1+2	speciality	compulsory	
Information engineering	3	2+1+2	speciality	compulsory	

Teaching Staff in Charge	Assoc.Prof. SOOS Anna, Ph.D., asoos@math.ubbcluj.ro
Lect. MICULA Sanda, Ph.D., smicula@math.ubbcluj.ro
Lect. OLAH-GAL Robert, Ph.D., robert.olah-gal@cs.ubbcluj.ro	
Aims	To acquire basic of Probability Theory and Mathematical Statistics, focusing on applications.	
Content	1. Field of events. Operations with events. Pprobability space: classical definition of 
   probability, axiomatic definition of probability. Conditional probability. 
   Independent events. Total probability formula, Bayes formula. Classical probabilistic 
   models(Bernoulli, Poisson, Pascal, Markov-Polya). 
2. Random variables and discrete laws of probability (binomial, hypergeometric, Poisson, 
   Pascal, geometric). Distribution function. Continuous random variables. Probability 
   density function. Continuous laws of probability (uniform, normal, Gamma,  
   exponential, &#967;2, Beta, Student, Cauchy). Independent random variables. Operations 
   with random variables.
3. Numerical caracteristics for random variables. Expectation. Variance. Covariance and  
   correlation coefficient. Moments (initial, central, absolut, factorial). 
   Median, quantile, quartile, mode, skewness, kurtosis. Inegalities (Chebyshev, 
   Hölder, Cauchy-Schwartz-Buniakovski, Liapunov).
4. Convergence in probability, almost surely convergence, convergence in distribution. 
   Law of large numbers: Markov theorem, Chebyshev theorem, Poisson theorem, Bernoulli 
   theorem. Limit theorems: Lindeberg theorem, Liapunov theorem, Lindeberg-Lévy theorem, 
   Moivre-Laplace theorem.
5. Descriptive statistics. Statistical distribution. Parameters of statistical 
   distribution.
6. Sampling theory. Samples. Sample functions. Sample mean. Sample variance. Sample 
   moment. Sample central moment. Sample distribution function.Glivenko theorem.   
   Kolmogorov theorem.
7. Estimation theory. Estimating functions. Absolutely correct  and correct 
   estimators. Fisher information. Rao-Cramer inequality. Methods of estimation: method 
   of moments, method of maximum likelihood, method of confidence intervals. Monte Carlo 
   method.
8. Testing statistical hypotheses. Critical region. Power of a test. Neyman-Pearson 
   lemma. Z-test and  T (Student)-test for the mean. &#967;2-test for variance. F-test for 
   ratio of variances. Tests for difference of  means. &#967;2-test for several  
   characterstics. &#967;2-test for contingence tables. Goodness-of-fit-tests: Kolmogorov-
   Smirnov, &#967;2.	
Bibliography	1. Agratini, O., Blaga, P., Coman, Gh., Lectures on Wavelets, Numerical Methods and 
   Statistics, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005.
2. Blaga, P., Calculul probabilităţilor şi statistică matemmatică. Vol. II. Curs şi 
   culegere de probleme, Universitatea "Babeş-Bolyai" Cluj-Napoca, 1994. 
3. Blaga, P., Statistică prin Matlab, Presa Universitară Clujeană, Cluj-Napoca, 2002.
4. Blaga, P., Rădulescu, M., Calculul probabilităţilor, Universitatea "Babeş-Bolyai" 
   Cluj-Napoca, 1987. 
5. Ciucu, G., Craiu, V., Inferenţă statistică, Editura Didactică şi Pedagogică, 
   Bucureşti, 1974.
6. Feller, W.,  An introduction to probability theory and its applications, Vol.I-II,   
   John Wiley, New York, 1957, 1966.
7. Iosifescu, M., Mihoc, Gh.,. Theodorescu, R., Teoria probabilităţilor şi statistică 
   matematică, Editura Tehnică, Bucureşti, 1966. 
8. Lisei, H., Probability theory, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2004.
9. Lisei, H., Micula, S., Soos, A., Probability Theory trough Problems and Applications, 
   Cluj University Press, 2006.
10. Shiryaev, A.N., Probability (2nd ed.), Springer, New York 1995.	
Assessment	The final grade will be computed as follows:
  - final written exam at the end of semester: 50%
  - evaluation at the seminar:                 20%
  - lab works  during the semester:            30% 
Students who wish to improve their written exam grade, can do so in the oral exam.