Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT
| MMP0003 | Probability Theory and Statistics |
| Teaching Staff in Charge |
Prof. BLAGA Petru, Ph.D., pblaga cs.ubbcluj.roAssoc.Prof. SOOS Anna, Ph.D., asoos math.ubbcluj.roLect. MICULA Sanda, Ph.D., smicula math.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, χ2, 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. χ2-test for variance. F-test for ratio of variances. Tests for difference of means. χ2-test for several characterstics. χ2-test for contingence tables. Goodness-of-fit-tests: Kolmogorov- Smirnov, χ2. |
| References |
|
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. |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |