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.

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%.

Evaluation:

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.

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.