Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Modelare şi simulare - în limba engleză
Programul de studii: Modelare si simulare - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MMC1007	Statistical Computational Models	

Semester	Hours: C+S+L	Category	Status	
1	2+2+0	specialty	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
BLAGA Petru	Ph.D.	Prof.	Numerical and Statistical Calculus	*	*		

III. Course objectives
Knowledge and simulation of some statistical models oriented on software products. Correct using of the mathematical statistical models. Knowledge some statistical software Development and construct of  procedures for statistical methods.	

IV. Course contents
1.	Basics of probability theory. 
2.	Sampling theory and estimation theory.
3.	Confidence intervals and testing statistical hypotheses for mean values and variances.
4.	Nonparametric and parametric goodness-of-fit tests.
5.	Multiple linear regression.
6.	Analyse of variance (ANOVA).	

V. Bibliography
1.	Agratini, O., Blaga, P., Coman, Gh., Lectures on Wavelets, Numerical Methods,   and Statistics, Casa Cărţii de Stiinţă, Cluj-Napoca, 2005.
2.	Blaga, P., Calculul probabilităţilor şi statistică matematică. Vol. II. Curs şi culegere de probleme, UBB, Cluj-Napoca, 1994.
3.	Blaga, P., Statistică matematică. Lucrări de laborator, UBB, Cluj-Napoca, 1999.
4.	Blaga, P., Statistica... prin Matlab, Presa Universitară Clujeană, Cluj-Napoca, 2002.
5.	Blaga, P., Mureşan, A. S., Matematici aplicate în economie, Vol. I, Transilvania Press, Cluj-Napoca, 1996.
6.	Blaga, P., Rădulescu, M., Calculul probabilităţilor, UBB, Cluj-Napoca, 1987.
7.	Iosifescu, M., Mihoc, Gh., Theodorescu, R., Teoria probabilităţilor şi statistica matematică, Editura Tehnică, Bucureşti, 1966.
8.	Mihoc, I., Calculul probabilităţilor şi statstică matematică, Part. I-II, UBB, Cluj-Napoca , 1994, 1995.
9.	Văduva, I., Analiză dispersională, Editura Tehnică, Bucureşti, 1977.	

VI. Thematic of didactic activities per weeks
•	Lecture 1 : Probability space. Random variables. Random vectors. Distribution function.  Probability density function. Conditional distribution function. Conditional probability density function. Numerical characteristics of random variables. Mean value. Variance.  Standard deviation. Covariance. Correlation coefficient.
      [6; Cap. 1,3, 4,5 ] ,  [4; Cap. 2],  [5; Cap. 2],  [1; Cap. 3]. 
•	Lecture 2 : Mean value and  covariance matrix of  random vector aleator. Conditional mean value. Conditional variance. Chebyshev inequality. Convegence in probability. Convergence in distribution (law). Weak law of large numbers. Limit theorems (Lindeberg-Lévy, Moivre-Laplace, corrections ofcontinuity). [6; Cap. 6,7,8,10,11] , [4; Cap. 2], [5; Cap. 2], [1; Cap. 3].
•	Lecture 3 : Sampling theory. Sample functions. Sample mean. Sample moment. Sample central moment. Sample variance. Sample distribution function. Glivenko theorem. Kolmogorov theorem.
      [2; Cap. 2] , [4; Cap. 4], [5; Cap. 3], [1; Cap. 3].
•	Lecture 4 : Estimation theory. Consistent estimator. Unbiased estimator. Absolutely correct estimator. Correct estimator. Likelihood function. Maximum likelihood method. Maximim likelihood estimator. Fisher information. Rao-Cramér inequality. Efficient estimator. Method of  confidence intervals. [2;Cap. 2] , [4 ;Cap. 5], [5 ;Cap. 3], [1 ;Cap. 3].
•	Lecture 5: Testing statistical hypotheses. Test for statistical hypothesis. Error of type I. Error  of type II. Power of a test. Z-test, T-test and confidence interval for mean value of a variable. χ2 – test and confidence interval  for variance of a variable. 
      [2; Cap. 2] ; [4; Cap. 5, 6], [5; Cap. 3], [1; Cap. 3].
•	Lecture 6 : Z-test, T-test and confidence intervals for difference of two mean values. F- test for ratio of two variances. [2; Cap. 2] , [4; Cap. 5, 6], [5; Cap. 3], [1; Cap. 3].
•	Lecture 7 :  Goodness-of-fit χ2  - test for multinomial distribution.  Nonparametric goodness-of-fit χ2  - test. Parametric goodness-of-fit χ2  - test. Homogeneity  χ2  - test. Independence χ2  - test. Goodness-of-fit Kolmogorov test. . Goodness-of-fit Kolmogorov-Smirnov test.
      [2; Cap. 2] , [4; Cap. 5, 6], [5; Cap. 3], [1; Cap. 3].
•	Lecture 8: Regression problem. Multiple linear model. Fitted least-squares multiple linear regression model. Multiple linear model with constant term. Coefficient of determination. Total variance equation.  [1; Cap. 3].
•	Lecture 9 : Gauss-Markov linear model. Gauss-Markov theorem. Unbiased estimators for coefficients. Unbiased estimator for variance. [1; Cap. 3].
•	Lecture 10 : Classical linear model. Probability law of the coefficient estimators. Probability law of the variance estimator. T-test for the coefficients of  model, confidence intervals for the coefficients of model. [1; Cap. 3].
•	Lecture 11 : Maximum likelihood estimators for coefficiens and variance.  Linear prediction problem. Estimator for prediction. Confidence interval for prediction. [1; Cap. 3].
•	Lecture 12 : F-test for all coefficients. F-test for a subset of coefficients. F-test for classical linear model with constant term. F-test for equality of some coefficients . F-test for identity of two linear models. ANOVA table.  [1; Cap. 3].
•	Lecture 13 : One-way analysis of variance. Total variance equation. F-test for equality of means of categories. ANOVA table. [1; Cap. 3].   
•	Lecture 14 : Analysis of variance with two and more factors. Two-way ANOVA without interaction. F-test for the null effect of a factor. Two-way ANOVA with interaction. F-test for the null effect of a factor. F-test for the null interaction effects. [1; Cap. 3].

•	Seminar 1: Laws of probability. Normal law.  χ2  law. Student law. F law. 
Homework 1: Present all distribution laws (discrete and continuous) recognized by Matlab.
•	Seminar 2 :  Multidimensional normal law. Other limit theorems. 
Homework 2: Give some Matlab procedures to present limit theorems.
•	Seminar 3 : Some properties of sample functions. Matlab procedures for sample functions.
Homework 3: Using of Matlab procedures for random generators.
•	Seminar 4 : Some examples of estimators.
Homework 4: Matlab procedures for estimators.
•	Seminar 5: Matlab procedures for Z-test and T-test. 
      Homework 5: Write some Matlab procedures for χ2 – test and confidence interval  for variance  
      of a variable. 
•	Seminar 6: Matlab procedures Z-test and T-test for difference of two mean values.  
Homework 6: Write Matlab procedures for F-test and confidence interval for ratio of  variances.
•	Seminar 7 : Matlab procedures for goodness-of-fit Kolmogorov test and goodness-of-fit   
      Kolmogorov-Smirnov test.
      Homework 7 : Write Matlab procedures for χ2 -test. 
•	Seminar 8: Simple linear model.
Homework 8: Matlab procedures for polynomal fitted.  
•	Seminar 9 :  Gauss-Markov simple linear model.
Homework 9: Matlab procedures for fitting of linear model.
•	Seminar 10: Classical simple linear model.
Homework 10: Matlab procedures for classical linear model.  
•	Seminar 11: Linear prediction for classical simple model.
      Homework 11: Writting and sending by e-mail of Homeworks 1-3 to evaluation.
•	Seminar 12 : Evaluation of  Homeworks 1-3.   
      Homework 12: Writting and sending by e-mail of Homeworks 4-7 to evaluation.
•	Seminar 13: Evaluation of  Homeworks 4-7.   
Homework 13: Writting and sending by e-mail of Homeworks 8-10 to evaluation.
•	Seminar 14: Evaluation of  Homeworks 8-10.   	

VII. Didactic methods used
Lectures, interactive presentations, exercises, individual study, homework assignments.	

VIII. Assessment
Final grade consists from:
•	Final writtenexam :                  50%
•	Activity during the semester : 25% 
•	Evaluation of homeworks :     25%
Technical details:
•	Writting and sending the solutions of  homeworks coming in the next week of proposing. The original solutions with interesting applications will be appreciated. 
•	Result, comment and evaluation of homework will be presented in the last three seminars of semester. 
•	Result of the written exam is given in the day of exam together the final grade. The contested  mark from written exam is analysed with each interested person.	

IX. Additional references
1.	Ermakov, S. M., Metoda Monte Carlo şi probleme înrudite, Editura Tehnică, Bucureşti, 1976.
2.	Lebart, L., Morineau, M. G., Fenelon, J.-P., Traitment des données statistiques, Dunod, Paris, 1982.
3.	Lehmann, E. L., Testing statistical hypotheses, Springer, New York, 1997.
4.	Montgomery, D. C., Peck, E. A., Vining, G. G., Introduction to linear analysis, John Wiley & Sons (3rd ed.), New York, 2001.
5.	Saporta, G., Probabilités, analyse des données et statistique, Édition Technip, Paris, 1990.
6.	Tassi, Ph., Methodes statistiques, Economica (2nd ed.), Paris, 1989.
7.	Schervish, M. J., Theory of statistics, Springer, New York, 1995.
8.	Stapleton, J. H., Linear statistical models, John Wiley & Sons, New York, 1995.
9.	Weerahndi, S., Exact statistical methods for data analysis, Springer, New York, 1994.	

    Decan,                                                                                 Şef de catedră,
    Prof. Dr. Leon Ţâmbulea					Prof. Dr. Petru Blaga

     Data, 15.09.2008						Titular curs,
								Prof. Dr.  BLAGA Petru