
Generating functions and their use in the theory of probability
Hujer, Peter ; Omelka, Marek (advisor) ; Bubelíny, Peter (referee)
Generating functions are suitable mathematical apparatus to describe the distribution of random variables. In this paper we introduce frequently used types of generating functions, their basic properties, uniqueness and benefits of use. We use this established mathematical apparatus for some known continuous and discrete distributions, but also the unusual practical examples. Then we fully utilize the strength of generating functions to solve interesting problems, often associated with creation of the theory of branching processes, an important area of probability theory.


Random walk
Baňasová, Barbora ; Omelka, Marek (advisor) ; Dostál, Petr (referee)
Random walk is a wellknown mathematical model used in various scientific fields. The aim of this thesis is to explain and to show the relation between the basic characteristics of simple random walk. The paper summarizes theoretical knowledge concerning this mathematical model in terms of its symmetrical or asymmetrical version. It deals with the derivation of absorbing probabilities, probability of the first and repeated return to origin and clasification of simple random walk states. The final part presents random walk in a wider perspective as a martingale. The conditions under which a random walk equals a martingale are established as well. It is also shown how it is possible to apply this more general mathematical structure on the model of random walk.

 

The Kelly Criterion
Kálosi, Szilárd ; Omelka, Marek (advisor) ; Hlávka, Zdeněk (referee)
The present work is devoted to the Kelly criterion, which is a simple method for choosing the amount of the bet for gambles with a positive expected value. In the first part of the work we introduce the mathematical explanation of the criterion, examine the capital after $n$ trials as a function of the bet, the longrun rate of return and asymptotical properties of the capital growth. In the second part we attempt to generalize the Kelly criterion from the first part for some other situations. Examples for a simple game and generalized situations illustrating the properties of the Kelly criterion and results from previous parts compose the last part of the work.


Kelly criterion in portfolio selection problems
Dorová, Bianka ; Kopa, Miloš (advisor) ; Omelka, Marek (referee)
In the present work we study portfolio optimization problems. Introduction is followed by chapter 2, where we introduce the concept of utility function and its relationship to the investor's risk attitude. To solve the optimization problem we consider the Markowitz portfolio optimization model and the Kelly criterion, which are recalled in the fourth and fifth chapter. The work also contains an extensive numerical study. Using the optimization software GAMS we solve portfolio optimization problems. We consider a portfolio problem with (and without) allowed short sales. We compare the obtained portfolios and we discuss whether Kelly optimal portfolio is a special case of the Markowitz optimal portfolio for the special value of the minimum expected return.


The Depth of Functional Data.
Nagy, Stanislav ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee)
The depth function (functional) is a modern nonparametric statistical analysis tool for (finitedimensional) data with lots of practical applications. In the present work we focus on the possibilities of the extension of the depth concept onto a functional data case. In the case of finitedimensional functional data the isomorphism between the functional space and the finitedimensional Euclidean space will be utilized in order to introduce the induced functional data depths. A theorem about induced depths' properties will be proven and on several examples the possibilities and restraints of it's practical applications will be shown. Moreover, we describe and demonstrate the advantages and disadvantages of the established depth functionals used in the literature (FraimanMuniz depths and band depths). In order to facilitate the outcoming drawbacks of known depths, we propose new, Kband depth based on the inference extension from continuous to smooth functions. Several important properties of the Kband depth will be derived. On a final supervised classification simulation study the reasonability of practical use of the new approach will be shown. As a conclusion, the computational complexity of all presented depth functionals will be compared.


Parameter Estimation under Twophase Stratified and Cluster Sampling
Šedová, Michaela ; Kulich, Michal (advisor) ; Picek, Jan (referee) ; Omelka, Marek (referee)
Title: Parameter Estimation under Twophase Stratified and Cluster Sampling Author: Mgr. Michaela Šedová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Mgr. Michal Kulich, Ph.D. Abstract: In this thesis we present methods of parameter estimation under twophase stratified and cluster sampling. In contrast to classical sampling theory, we do not deal with finite population parameters, but focus on model parameter inference, where the ob servations in a population are considered to be realisations of a random variable. However, we consider the sampling schemes used, and thus we incorporate much of survey sampling theory. Therefore, the presented methods of the parameter estimation can be understood as a combination of the two approaches. For both sampling schemes, we deal with the concept where the population is considered to be the firstphase sample, from which a sub sample is drawn in the second phase. The target variable is then observed only for the subsampled subjects. We present the mean value estimation, including the statistical prop erties of the estimator, and show how this estimation can be improved if some auxiliary information, correlated with the target variable, is observed for the whole population. We extend the method to the regression problem....

 
 
 