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

SYLLABUS

I. General data
Code	Subject	
MMC1007	Computational Statistical Models	

Semester	Hours: C+S+L	Category	Status	
1	2+1+0	speciality	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.	Applied Mathematics	*	*		

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.  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.
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 of continuity). 
3. Sampling theory. Sample functions. Sample mean. Sample moment. Sample central moment. Sample variance. Sample distribution function. Glivenko theorem. Kolmogorov theorem.
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. 
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. 
6. Z-test, T-test and confidence intervals for difference of two mean values. F- test for ratio of two variances.
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.
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.
9. Gauss-Markov linear model. Gauss-Markov theorem. Unbiased estimators for coefficients.   Unbiased estimator for variance.
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.
11. Maximum likelihood estimators for coefficients and variance.  Linear prediction problem. Estimator for prediction. Confidence interval for prediction.
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.
13. One-way analysis of variance. Total variance equation. F-test for equality of means of  categories. ANOVA table.
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.	

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 varaince. Chebyshev inequality. Convegence in probability. Convergence in distribution (law). Weak law of large numbers. Limit theorems (Lindeberg-Lévy, Moivre-Laplace, corrections of continuity). [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 coefficients 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. 	

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

VIII. Assessment
Final grade consists from:
•	Final written exam:                 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. 
•	Results 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.	

	Dean,							Chair head,

	Date,	September  27, 2009				Course responsible,
								Prof. BLAGA Petru, Ph.D.