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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 5 Name and surname MICULA Sanda Name and surname Hours: C+S+L 2+1+2 Category specialty Status mandatory II. Full status faculty members Scientific Didactic Chair title title Ph.D. Lect. Numerical and Statistical Calculus Associated faculty members Scientific Institution title Type of position Type of activity C * S * Type of activity C S 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. 1. 2. 3. 4. 5. 6. 7. 8. L * IV. Course contents Field of events. Probability. Classical probabilistic models. Random variables and random vectors. Probability distributions. Numerical characteristics of random variables. Sequences of random variables. Laws of large numbers. Limit theorems. Descriptive statistics. Sample theory. Estimation theory. Hypothesis and significance testing. V. Bibliography 1. Agratini, O., Blaga, P., Coman, Gh., Lectures on Wavelets, Numerical Methods and Statistics, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005. L 2. Blaga, P., Calculul probabilităţilor şi statistică mateamtică. 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. 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. Bertrand’s paradox. 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. 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: Cumulative distribution function. Inverse cumulative distribution function. Random number generators. 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: Numerical characteristics of random variables. 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: Sequences of random variables. Laws of large numbers. 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: Graphical display of data. 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: Calculative descriptive statistics. Glivenko’s theorem. 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 RaoBlackwell 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 Chisquared-test for contingency tables. Lab 14: Completion of lab works. Review. VII. Learning 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: 20% - lab works during the semester: 30% Students who wish to improve their written exam grade can do so in the oral exam. Details about evaluation Evaluation in seminar: consists of attendance, class participation, homework, bonus problems. Seminars will be held for the first seven weeks (two hours per week) and in the 8th week there will be a written midterm exam consisting of problems similar to the ones discussed in seminar. All these will produce a grade from 1 to 10 for evaluation in seminar. All laboratories are mandatory, each lab being due the week following the week when it was assigned; each lab will be graded from 1 to 10; there is a “grace period” of one week for each lab, meaning that it can be turned in one week late, for a lower grade; labs that are not turned in within two weeks of being assigned are not accepted anymore, being graded with a 1. Evaluation for lab works will consist of the average of the grades of all the labs. The results of the written exam will be made known 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, ClujNapoca, 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. Date, April 25, 2007 Course responsible, Lect. MICULA Sanda, Ph.D.