Probability Theory

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

Last updated: 2019. október 10.

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

Computer Engineering BSc

Course ID Semester Assessment Credit Tantárgyfélév
VISZAB02   2/2/0/v 5  
3. Course coordinator and department Dr. Csákány Rita,
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

Elementary combinatory, calculus

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

ÉS NEM (TárgyEredmény("BMEVISZA208", "jegy", _) >= 2
TárgyEredmény("BMEVISZA208", "felvétel", AktualisFelev()) > 0
TárgyEredmény("BMEVISZAB00", "jegy", _) >= 2 )
TárgyEredmény("BMEVISZAB00", "felvétel", AktualisFelev()) > 0
TárgyEredmény( "BMEVISZAB00" , "aláírás" , _ ) = -1 )

ÉS (Training.Code=("5N-A8") VAGY Training.Code=("5NAA8"))

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

A kötelező előtanulmányi rendek grafikus formában itt láthatók.

7. Objectives, learning outcomes and obtained knowledge Learn 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 hours lectures and 2 hours practices/week


10. Assessment

Period of study:


Participation in the exercises is compulsory (TVSZ § 14 (3) in Code of studies and exams).
During the semester, a 120-point test run will be written, which is sufficient to fulfill this at least sufficient (40 points).


Exam period:


The exam is written. The results of the semester interim test result and the examination result are included in the test result at ratio 40% -60%.


If the total score is        40-54 points sufficient (2),

55-69 points medium (3),

70-84 points good (4),

85-120 marks (5).

In case of written exam’s result at least sufficent, it is possible to change 1 mark upwards and downwards, depending on the oral answer.

11. Recaps



The repetition option for the midterm test: the inadequate result can be improved or the missing result can be replaced in a retake test. If the student has not written a midterm test, writing the retake midterm test is compulsory. Without writing this he or she will not get signature from the subject. If someone has failed the midterm test, but does not participate in the substitution occasion, can try to get the signature for a replacement fee in the first week of the exam period.


At the retake midterm test it is also possible to try to improve the reached results in case of a successful midterm test’s result. The valid result will be the greater one among the new result and 40.The new score will be valid even if it is worse than the original. An valid result already cannot be improved in the exam period.


12. Consultations

During the semester at the lecturer's scheduled reception hours. We organize consultations before midterm tests and exams.

13. References, textbooks and resources


  1. Ketskeméty László: Valószínűségszámítás, Műegyetem Kiadó 55050
  2. Ketskeméty László, Pintér Márta (szerk.):Valószínűségszámítás feladatgyűjtemény megoldásokkal, Arteria Studio Kiadó, 2011
  3. Vetier András elektronikus jegyzete:
14. Required learning hours and assignment
Preparation for lectures 8
 Preparation for practices 20
Preparation for tests
Preparation for exams
15. Syllabus prepared by




Dr. László Ketskeméty

associate professor

Department of Computer Science and Information Theory

IMSc program

A.) (High-level thematic and methodological elements) On the practices we start discussing theoretical backgrounds, where we prove such theorems that were not included in the lectures.

B.) (Exceptions to Regular Training Practices) There are fewer simple example types, more complex, tricky exercises will discussed on the exercises. The typical examples are available on internet. The IMSc students can practice on them at home.

C.) Students receive homeworks for extra IMSC points.
IMSc score

During the semester each student can get maximum 25 IMSC points from probability theory.


Students can get 7 points for the homeworks and for the activities in the practices. This is calculated below. Denote hw the percent of the performed homework's points. 


1 point if 1≤hw≤21;

2 points if 21≤hw≤40;

3 points if  41≤hw≤60

4 points if 61≤hw≤80

5 points if 81≤hw≤100


The value obtained can be increased by the practice leader 2 if the student was active on the exercises.


Moreover a student can get other 9-9 extra IMSC points for midterm test and exam test, respectively. Its calculation algorithm is as follows. A student can get IMSC points for the midterm test (or exam test) if the score is at least 90 out of the 120 points available. If zh denotes the student's accomplished result, the IMSC points are calculated as follows:


1 point if 90≤zh≤94;

2 points if 95≤zh≤99;

3 points if 100≤zh≤102;

4 points if 103≤zh≤105;

5 points if 106≤zh≤108;

6 points if 109 ≤zh≤110;

7 points if 111 ≤zh≤113;

8 points if 114 ≤zh≤116;

9 points if 117 ≤zh≤120.


A total of 25 IMSC points can be obtained through the semester. The points will be valid if the exam result at least passed (2). The acquisition of the IMSC points is also provided for non-IMSC students.