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

SUBJECT

Code
Subject
MC268 Mathematical Statistics with Applications
Section
Semester
Hours: C+S+L
Category
Type
Applied Mathematics
2
2+1+1
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 product (Matlab), with applications in ecomomics, medicine, etc.
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 variance. 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 coefficiens 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, O., COMAN, Gh.: Lectures on Wavelets, Numerical Methods and
Statistics, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005.
2. BLAGA, P.: Statistică... prin Matlab, Presa Universitară Clujeană, Cluj-Napoca, 2002.
3. LEBART, L. - MORINEAU, M.G. - FÉNELON, J.-P.: Traitment des données statistiques.
Paris: Dunod, 1982
4. LEHMANN, E.L.: Testing statistical hypotheses. New York: Springer, 1997.
5. MONTGOMERY, D.C. - PECK, E.A. - VINING, G.G.: Introduction to linear analysis. New
York: John Wiley & Sons (3rd ed.), 2001.
6. SAPORTA, G.: Probabilités, analyse des données et statistique. Paris: Editions
Technip, 1990.
7. TASSI, Ph.: Methodes statistiques. Paris: Economica (2nd ed.), 1989.
8. SCHERVISH, M.J.: Theory of statistics. New York: Springer, 1995.
9. STAPLETON, J.H.: Linear statistical models. New York: John Wiley & Sons, 1995.
10.WEERAHANDI, S.: Exact statistical methods for data analysis. New York: Springer, 1994.
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