Probability Theory

A tantárgy neve magyarul / Name of the subject in Hungarian: Valószínűségszámítás

Last updated: 2015. november 26.

Budapest University of Technology and Economics
Faculty of Electrical Engineering and Informatics

Budapest University of Technology and Economics, Faculty of Electrical Engineering and Informatics

B.Sc. in Engineering Information Technology

Course ID Semester Assessment Credit Tantárgyfélév
VISZAB00 3 2/2/0/v 4  
3. Course coordinator and department Csehi Csongor György,
Web page of the course
4. Instructors

László Ketskeméty associate professor Department of Computer Science and Information Theory
Márta Barbara Pintér associate professor Department of Computer Science and Information Theory

5. Required knowledge


6. Pre-requisites
((TárgyEredmény( "BMETE90AX21" , "jegy" , _ ) >= 2
TárgyEredmény( "BMETE90AX04" , "jegy" , _ ) >= 2)

ÉS NEM (TárgyEredmény("BMEVISZA208", "jegy", _) >= 2
TárgyEredmény("BMEVISZA208", "felvétel", AktualisFelev()) > 0
TárgyEredmény("BMEVISZAB02", "jegy", _) >= 2 )
TárgyEredmény("BMEVISZAB02", "felvétel", AktualisFelev()) > 0
TárgyEredmény( "BMEVISZAB02" , "aláírás" , _ ) = -1 )

A fenti forma a Neptun sajátja, ezen technikai okokból nem változtattunk.

A kötelező előtanulmányi rendek grafikus formában itt láthatók.


The studets must perfom subjects BMETE90AX21 and BMEVISZA208

7. Objectives, learning outcomes and obtained knowledge

Learning the basics of stochastic modeling

8. Synopsis
  1. Historical introduction. Basic concepts: random experiment, event space, event, elementary event, operation between events, axioms, sigma algebra

  2. Properties of probability: Poincare-rule, Boole’s inequalities, continuity of probability

  3. Conditonial probability, independency of events, Theorem of total probability, Bayes’s Theorem, produce theory

  4. Classical probability, geometrical probability. Examples for application: urn models, Buffon’s needle

  5. Random variable, probability distribution function, discrete and continouos cases, properties of the distribution function, probability of falling in an interval, discrete distribution, probability density function

  6. Notable discrete random variables: binomial, Poisson, geometrical.  Poisson approximation to the binomial distribution. Memoryless properties of the geometric distribution

  7. Notable continouos distributions: uniform, exponential, normal. Simulation with uniform distribution. Memoriless property of the exponential distribution. Standard normal distribution. Linear transformation

  8. Expected value, deviation, moments.  Theorems for expected value and deviation. Expected value and deviation of notable dispersions.

  9. Steiner’s Theorem, Markov’s- and Chebisev’s inequalities.

  10. Joint distribution function, projective distribution functions. Independency, convolution (discrete and continouos cases), Joint density function, projective density function

  11. Theorems of large numbers: Weak- and Stronge Law of Large Numbers. Central limit theorems, Moivre-Laplace’s Theorem

  12. Covariance, correlation. Properties of covariance and correlations. Connection between independency and uncorrelatedness

  13. Conditional distribution, conditional expectation (regression). Linear regression. Properties of regression. Examples of discrete and continouos cases

  14. Two-dimensional normal distribution,  polynomial distribution. Connection of the independency and uncorrelatedness in normal case. Projections of the polinomial distribution are binomials

9. Method of instruction

2 lectures +2 practises per week

10. Assessment



mandatory participation in the practices (TVSZ 14§ (3)).

During the semester two mid-term tests it will take place, at least sufficient to fulfill them individually (40%) of the signature condition.




The exam is written. The results of the examination paper and midterm tests 50% to 50% counted in mark of the exam. Evaluation:
the total score of 40% -54%: 2
55% -69% : 3,
70% -84%: 4,
85% to 100%: 5th
In case of at least two mark for  exam paper, it is possible to change in one mark in oral exam.

11. Recaps

One replacement option belong for every midterm tests. It can be replaced in the case of failed test or the absences. Only one of the two midterm tests can be replaced. Everyone can try to write again the unsuccessful paper on the signature supplementary exam.

12. Consultations

On the lecturer’s office time the students can keep contact with the teacher.  Consultations are organized before every exams.

13. References, textbooks and resources

Grinstead, C. M.- Snell, J. L. : Introduction to Probability, American Mathematical Society, ISBN: 0821807498 (

14. Required learning hours and assignment
Kontakt óra56
Félévközi készülés órákra22
Felkészülés zárthelyire22
Házi feladat elkészítése
Kijelölt írásos tananyag elsajátítása
15. Syllabus prepared by

László Ketskeméty