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Unfair Ballots
Valášková, Zuzana ; Hlubinka, Daniel (advisor) ; Lachout, Petr (referee)
OBSAH Nazov prace: Nespravodlivc losovanie Antor: Zuzana Valaskova Katedra: Katedra pravdepodobnosti a matematicke statistiky Veduci bakalarskej prace: R.NDr.Daniel Hlnbinka, Ph.D. e-mail veduceho pracc: hluhu]k;i:'"Jkarlm.mif.cuni.cz Abstrakt: V predlozencj praci studujem otazku ncspravodliveho losovania, ktorn som sku- mala prostrednictvom silneho nastroja, testovania hypotez. Na zaciatku som sa zaoberala najdenim vhodnej nahodnej veliciny. ktora by iiam umoznila jedno- duchym sposobom rozrieait! povodnu otazkn spravodlivosti losovania. Zaoberala som sa otazkou randomizovanych a nerandomizovanych testov. V praci su okrein ineho nvedenc iri vzorove priklady s podrobnyrn riescnim. Vo vsetkych jc nvedena aplikacia randoinizovaneho i nerondomizovaneho testu. Prvy ilustruje prave take losovaiiie, u ktoreho neniame dostatocne dovody na pochybovanie o spravodlivosti celeho losovania, v drnhoni priklade je uvedeny typicky pri[)ad ncspravodliveho losovania a v l.rel'oTn je uvedeny pripa.fi. kedy t,(^sty nedavaju rovnake vyskxlky. Prilohu tvori zdrojovy text nnmerickych vypoctov v Mathernatice. Kmmve slova: losovanie, sf>rav(>dlivost'.tnst Title: Unfair Ballots Author: Zuzana Valaskova Department: Department of Probability and Mathematical Statistics Supervisor: RNDr.Daniel Illubinka, Ph.D. Supervisor's e-mail...
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Itôův a Stratonovičův stochastický integrál
Voldán, Adam ; Hlubinka, Daniel (advisor) ; Dostál, Luboš (referee)
In this thesis the Ito stochastic integral and the Stratonovich stochastic integrals are studied. Their basic and some special properties are shown. Further the theory of the numerical solution of stochastic differential equations (SDE) is introduced. Using simple examples the properties of chosen numerical schemes are presented. Finally the Black-Scholes-Merton formula for pricing of European call option is sketched, and similar problems are numerically solved using the above presented algorithms.
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Selected topics of random walks
Filipová, Anna ; Hlubinka, Daniel (advisor) ; Beneš, Viktor (referee)
The theme of this thesis are symmetric random walks. We define different types of paths and prove the reflection principle. Then, based on the paths, we define random walks. The thesis also deals with probabilities of returns to the origin and first returns to the origin, further with probabilities of number of changes of sign or returns to the origin up to a certain time. We also define the maximum of the random walk and the first passage through a certain point. In the second chapter, we solve several problems, which form the proofs of some theorems from the first chapter or complement the first chapter in a different way. For example, we prove geometrically that the number of paths of one type equals the number of paths of another type or we compute the probability that there occurs a certain number of changes of sign up to a given time.
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Depth of variance matrices
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
The scatter halfspace depth is a quite recently established concept which extends the idea of the location halfspace depth for positive definite matrices. It provides an interest- ing insight into the problem of suitability quantification of a matrix for the description of the covariance structure of the multivariate distribution. The thesis focuses on the investigation of theoretical properties of the depth for both general and more specific probability distributions which can be used for data analysis. It turns out that the es- timators of scatter parameters based on the empirical scatter depth are quite effective even under relatively weak assumptions. These estimators are useful especially for dealing with a sample containing outliers or contaminating observations. 1
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Multivariate distributions in Cartesian, polar and directional coordinates
Bečková, Magdaléna ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
The thesis focuses on the distributions of random vectors in Cartesian, polar and directional coordinates. In the thesis we derive formulas for probability density func- tions of two-dimensional vectors in polar and directional coordinates, three-dimensional vectors in spherical and directional coordinates and n-dimensional vectors in spherical coordinates. These formulas are shown on several examples of normal and uniform distri- butions. Finally, the thesis discusses differences between the probability density functions in particular coordinates systems. 1
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Probability distribution of functional random variables
Dolník, Viktor ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
We describe basic notions of functional random elements and the space of functions L2 [0, 1]. We discuss the non-existence of a probability density functional and the re- quirements for integrating in a functional space. In Chapter 2, we define distribution functionals and introduce a goodness-of-fit test which utilises them. The concept of char- acteristic functionals follows in Chapter 3, along with the latest test for Gaussianity of functional random elements. We conclude the chapter with our own new goodness-of- fit test, where we prove the distribution of its test statistic under the alternative, then under the null hypothesis, and lastly the distribution of the bootstrapped test statistic. Finally, we illustrate the theory on a simulation study of the empirical significance level and power of the goodness-of-fit tests. 1
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Selected properties of bivariate and multivariate random walks.
Nguyen, Huy Quang ; Hlubinka, Daniel (advisor) ; Kříž, Pavel (referee)
This thesis deals with random walks with emphasis on multivariate random walks. We focus mainly on return of the random walk to the origin in two dimensions. Some results are generalized in any dimension. Specifically we discuss the probability of return to the origin, probability of the first return to the origin and expected time of the first return. In the thesis we also find the arcsine laws and short simulation study focused on multivariate version of this topic. 1
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