## Babes-Bolyai University of Cluj-Napoca Faculty of Mathematics and Computer Science Study Cycle: Master SUBJECT

 Code Subject MMC1003 Mathematical Statistics with Applications
 Section Semester Hours: C+S+L Category Type Applied Mathematics 2 2+2+0 speciality compulsory
 Teaching Staff in Charge
 Prof. BLAGA Petru, Ph.D.,  pblagacs.ubbcluj.ro
 Aims Knowledge of some moderns methods of mathematical statistics oriented to soft products.
 Content • 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. • 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). • Sampling theory. Sample functions. Sample mean. Sample moment. Sample central moment. Sample variance. Sample distribution function. Glivenko theorem. Kolmogorov theorem. • 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. • 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. • Z-test, T-test and confidence intervals for difference of two mean values. F- test for ratio of two variances. • 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. • Regression problem. Multiple linear model. Fitted least-squares multiple linear regression model. Multiple linear model with constant term. Coefficient of determination. Total variance equation. • Gauss-Markov linear model. Gauss-Markov theorem. Unbiased estimators for coefficients. Unbiased estimator for variance. • 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. • Maximum likelihood estimators for coefficients and variance. Linear prediction problem. Estimator for prediction. Confidence interval for prediction. • 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. • One-way analysis of variance. Total variance equation. F-test for equality of means of categories. ANOVA table. • 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.
 References 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.
 Assessment Final grade consists from: • Final written exam : 50% • Activity during the semester : 25% • Evaluation of homeworks : 25%
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject