Download MMP0003 Probability Theory and Statistics

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.
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2.
3.
4.
5.
6.
7.
8.
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*
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.