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

SYLLABUS

I. General data
Code	Subject	
MMP0003	Probability Theory and Statistics	

Semester	Hours: C+S+L	Category	Status	
3	2+1+2	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
MICULA Sanda	Ph.D.	Lect.	Applied Mathematics	*	*	*	

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
To acquire basic knowledge of Probability Theory and Mathematical Statistics, with main focus on applications. To become familiar and be able to work with various probabilistic and statistical models. To be able to perform statistical analysis of data.	

IV. Course contents
1.	Field of events. Probability. Classical probabilistic models.
2.	Random variables and random vectors. Probability distributions. 
3.	Numerical characteristics of random variables. 
4.	Sequences of random variables. Laws of large numbers. Limit theorems.
5.	Descriptive statistics. 
6.	Sample theory. 
7.	Estimation theory.
6.   Hypothesis and significance testing.   	

V. Bibliography
1. Micula, S., Probability and Statistics for Computational Sciences, Cluj University Press, 2009.
2. Agratini, O., Blaga, P., Coman, Gh., Lectures on Wavelets, Numerical Methods and 
   Statistics, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005.
3. Blaga, P., Calculul probabilităţilor şi statistică matemmatică. Vol. II. Curs şi 
   culegere de probleme, Universitatea "Babeş-Bolyai" Cluj-Napoca, 1994. 
4. Blaga, P., Statistică prin Matlab, Presa Universitară Clujeană, Cluj-Napoca, 2002.
5. Blaga, P., Rădulescu, M., Calculul probabilităţilor, Universitatea "Babeş-Bolyai" 
   Cluj-Napoca, 1987. 
6. Ciucu, G., Craiu, V., Inferenţă statistică, Editura Didactică şi Pedagogică, 
   Bucureşti, 1974.
7. Feller, W.,  An introduction to probability theory and its applications, Vol.I-II,   
   John Wiley, New York, 1957, 1966.
8. Iosifescu, M., Mihoc, Gh.,. Theodorescu, R., Teoria probabilităţilor şi statistică 
   matematică, Editura Tehnică, Bucureşti, 1966. 
9. Lisei, H., Micula, S., Soos, A., Probability Theory trough Problems and Applications, 
   Cluj University Press, 2006.
10. Milton, J.S., Arnold, J. C., Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences, 3rd Edition. McGraw-Hill, New York, 1995.
11. Shiryaev, A.N., Probability (2nd ed.), Springer, New York 1995.	

VI. Thematic of didactic activities per weeks
Lecture 1: Experiments, events, field of events, operations with events. Classical definition of probability. Axiomatic definition of probability. Poincaré’s formula. Geometric probability. Buffon’s needle problem. 
Seminar 1: Euler’s Gamma and Beta functions. Properties.
Lab 1: Introduction to Matlab, I.
Lecture 2: Conditional probability. Independent events. Total probability formula, Bayes’ formula. Classical probabilistic models (binomial, multinomial, hypergeometric, Poisson, Pascal, geometric). 
Seminar 2: Classical probability problems. Geometric probability. Conditional probability. Independent events. Bayes’ formula.
Lab2: Introduction to Matlab, II.
Lecture 3: Random variables and random vectors. Discrete random variables. Probability distribution function. Cumulative distribution function. Discrete probability laws (Bernoulli, binomial, hypergeometric, Poisson, Pascal, geometric). Functions of discrete random variables.
Seminar 3: Classical probabilistic models.
Lab 3: Discrete random variables. Probability distribution function. 
Lecture 4. Continuous random variables. Probability density function. Continuous random variables. Probability density function Continuous probability laws (uniform, normal, Gamma, exponential, Chi-squared, Beta, Student, Cauchy, Fischer). Independent random variables. Functions of continuous random variables.
Seminar 4: Discrete random variables. Functions of discrete random variables.
Lab 4: Continuous random variables. Probability density function. CDF and Inverse CDF.
Lecture 5. Numerical characteristics of random variables. Expectation. Variance. Moments (initial, central, absolute). Covariance and correlation coefficient. Quantile, median, quartiles. Inequalities (Hölder, Schwartz, Cauchy-Buniakovski, Minkowsky, Markov, Chebyshev, Kolmogorov).
Seminar 5: Continuous random variables. Functions of continuous random variables. Random vectors.
Lab 5: Numerical characteristics of random variables.
Lecture 6. Sequences of random variables. Convergence in probability, almost surely convergence, convergence in distribution. Laws of large numbers. The theorems of Hincin, Markov, Chebyshev, Poisson, Kolmogorov.
Seminar 6: Numerical characteristics of random variables. Inequalities.
Lab 6: PDF and CDF of continuous distributions. Random number generators.
Lecture 7: Limit theorems. The theorems of Lindeberg, Liapunov, Lindeberg-Lévy, Moivre-Laplace. Other limit theorems.
Seminar 7: Sequences of random variables. Laws of large numbers.
Lab 7: Descriptive Statistics. Grouped frequency distribution, graphical display of data.
Lecture 8. Descriptive statistics. Data collection. Graphical display of data. Frequency distribution and histograms. Parameters of a statistical distribution. Measures of central tendency. Measures of variation. Correlation and regression. Linear regression.
Lab 8: Correlation and regression. 
Lecture 9. Sample theory. Samples. Sample functions. Sample mean. Sample variance. Sample moments. Sample central moments. Sample distribution function. Glivenko’s theorem. Kolmogorov’s theorem.
Lab 9: Confidence intervals  for large samples.
Lecture 10. Estimation theory. Point estimators. Unbiased estimators. Confidence intervals. 
Lab 10: Confidence intervals concerning means.
Lecture 11: Properties of point estimators. Sufficient statistics. Likelihood function.  The Rao-Blackwell theorem and minimum variance estimators. Fisher’s information. Absolutely correct estimators. The Rao-Cramer inequality. 
Lab 11: Confidence intervals concerning variances.
Lecture 12: Methods of estimation. The method of moments estimator, the method of maximum likelihood estimator.
Lab 12: Hypothesis and significance testing concerning means.
Lecture 13: Hypothesis testing. Rejection region. Type I errors. Significance testing and P-values. The Z-test and T (Student)-test for the mean. The Chi-squared-test for variance. The F-test for the ratio of variances. Tests for the difference of means. 
Lab 13: Hypothesis and significance testing concerning variances.
Lecture 14: Type II errors and the power of a test. The most powerful test and the Neyman-Pearson lemma. Uniformly most powerful tests. The Chi-squared-test for several characteristics. The Chi-squared-test for contingency tables.
Lab 14: Lab exam.	

VII. Didactic methods used
Lecture, description, explanation, conversation, discussion, examples, synthesis, group-based work, individual study, homework. Group discussion over theoretical issues, as well as applications and modeling of the notions and methods presented.	

VIII. Assessment
The final grade will be computed as follows:
  - final written exam at the end of the semester: 50%
  - evaluation at the seminar:                                25%
  - evaluation of lab works:                                  25% 
Details about evaluation 
•	Evaluation in seminar: consists of attendance, class participation, homework, bonus problems.
      All these will produce a grade from 1 to 10 for evaluation in seminar.
•	Each week there will be a lab assignment, closely related to the topics presented at the lecture that week. The assignment will be discussed in class. Students can work in groups or individually. Lecture and seminar notes will be necessary for completion of the lab assignment. 
•	Evaluation for lab works will consist of attendance, class participation and a lab exam (held during the last week of classes). All these will produce a grade from 1 to 10 for evaluation of lab works.
•	The results of the written exam will be announced the day of the exam, as well as the final grade.	

IX. Additional references
1.	Blaga, P., Mureşan, A. S., Matematici aplicate în economie, Vol. I, Transilvania Press, Cluj-Napoca, 1996.
2.	Feller, W.,  An introduction to probability theory and its applications, Vol.I-II, John Wiley, New York, 1957, 1966.
3.	Gnedenko, B. V., The theory of probability and the elements of statistics (Fifth ed.), AMS Chelsea Publishing, Providence, RI, 2005. 
4.	Lebart, L., Morineau, M. G., Fenelon, J.-P., Traitment des données statistiques, Dunod, Paris, 1982
5.	Lehmann, E. L., Testing statistical hypotheses, Springer, New York, 1997
6.	Saporta, G., Probabilités, analyse des données et statistique, Édition Technip, Paris, 1990
7.	Tassi, Ph., Méthodes statistiques, Economica (2n ed.), Paris, 1989
8.	Schverish, M. J., Theory of statistics, Springer, New York, 1995
Shiryaev, A.N., Probability (2nd ed.), Springer, New York 1995.	

	Dean,							Chair Head,
            Prof. Dr. Leon Ţâmbulea                                          Prof. Dr. Octavian Agratini

  
            Date,							Course responsible,
	November 29, 2009				        Lect. MICULA Sanda, Ph.D.