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

SUBJECT

Code
Subject
MMP0003 Probability Theory and Statistics
Section
Semester
Hours: C+S+L
Category
Type
Computer Science
5
2+1+2
speciality
compulsory
Teaching Staff in Charge
Prof. BLAGA Petru, Ph.D.,  pblagacs.ubbcluj.ro
Assoc.Prof. SOOS Anna, Ph.D.,  asoosmath.ubbcluj.ro
Lect. MICULA Sanda, Ph.D.,  smiculamath.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