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Urn models with stochastic replacements
Kochaniková, Petra ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
The thesis purpose is to discuss urn models where the probability of success at any trial depends upon the number of previous successes. Such a scheme allows us to estimate the number of HIV cases among intravenous drug users. The coef- ficients in known probability generating function will be derived for the number of new infectives generated in both homogenous and inhomogenous population. The expectations and variances of the number of new infectives are also derived for both cases. These derived values will be verified for some fixed number of infectives and susceptibles by simulations. In the end of this thesis the studied model will be applied on a practical example where the effect of vaccination will be studied. 1
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Third order moment characteristics for spatial point processes
Verchière, Didier ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
Moment characteristics are widely used for the statistical analysis of spatial point processes. Standard summary statistics used for the analysis of point processes are of first and second order (intensity, K -function, pair-correlation function...). Nonetheless, none of these characteristics describes the distribution of a point pattern completely. Higher order characteristics such as third-order characteristics can give more information about the spatial interactions. Two such characteristics have already been studied: the z -function (Moller et. al. 98) and the T -function (Schladitz, Baddeley 2000). Key words: T -function, z -function , third order moment characteristics.
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Bivariate distributions
Bednárik, Vojtěch ; Pawlas, Zbyněk (advisor) ; Klebanov, Lev (referee)
The thesis deals with three selected constructions of bivariate distributions. The first approach is to use the Fréchet bounds, which determine restrictions on the distribution function and the correlation coefficient of bivariate distribution. The second construction is the Plackett distribution which is a class of distributions containing the Fréchet bounds and the member corresponding to independent random variables. The third construction is a trivariate reduction method that is used for a construction of bivariate gamma, exponen- tial and Poisson distribution. Only bivariate Dirichlet distribution has slightly different construction. For the last four mentioned distributions the following basic characteris- tics are derived: density function, marginal distributions, correlation coefficient and some conditional moments, in case of exponential and Dirichlet distribution even conditional distribution. 1
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Firm Valuation
Baran, Jaroslav ; Pawlas, Zbyněk (advisor) ; Hurt, Jan (referee)
Xazev prace: Ocenovani pndnikii Aulor: Jaroslav Ha ran Katedra (iistav): Katedra pravdepodobnost i a mateniat icke statisliky Vedouci bakalarske pra.ce: HNDr. Zbynek Pawlas. Ph.D. e-mail vedoucfho: pawlas'.^'karlin.mtf.cnni.c/, Abslrakt: C'ileni prace je seznamif c.tenafe sc zfddadnhni a nejpon/iVanejsimi nastroji pro ohodnoceni akcii akciovycli spolecnosti. Druha cast. pra.ce so zabyva nitr'todaini odvoxt'iii a odliadu pai'ainetru pou/itych v modnlccli ororiovani. Jsiiu pr(i/('iit,ovaiiy zakladnf niodcly otvnnvani jako in^lnda sourasnr hodnoly budouci'cli prijniii a rclalivtii Dci-fiuvani, ktcrr \wr v uvahu tr/ui ccny srovnatelnych firciti. N'a konci pnicc jc jirakticky pri'klad occnrni spulrrnusti Nukia. Prace ina ypis in- Ibriuativiii chara.kU'i' a, ji-.jnn cilnn uoni prusadil konkirl.ni morlcl. ale srznainit, ctcnafc sc '/iikladiiiini prinripy ixTfiovani a poukaxat na vyhucly a ncvyhody uvc- drnych inodcln. Investor se nakoiiec nefidi ponze ]>odle niodehi (K-in'iovani. ale /a- iijine poatoj i diky oritatiiini /drojiiin inlorniaci. jako jsou media, vyrorni zpravy spcileeiiostf. /.pravy analytiku. vylih'dky trim atd. Klirova slova: ocenovani. soneasna liodnota, rust, urokova infra Title: Irinu valuation Author: .laroslav I'ia.ran De]jarlment: Depa,rlinent ol Proba.bility and Ma.thcnia,! ical Staiistics...
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Stochastic models for neural spike trains
Vörösová, Estera ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
K modelovaniu prenášaní správ v nervovom systéme sa dajú využi' časové bodové procesy. Cie©om práce je popísa' vybrané typy bodových procesov, kon- krétne: Poissonov proces, proces obnovy a Coxov proces. "alej analyzujeme reálne dáta, testujeme vhodnos' jednotlivých pravdepodobnostných modelov. Najprv sa zoznámime s históriou skúmania nervových impulzov ako bodových procesov. V prvej kapitole sú zhrnuté neurofyziologické základy fungovania neurónov. V dru- hej časti pozornos' je venovaná popise vybraných bodových procesov a v poslednej kapitole vyberieme model a testujeme jeho vhodnos' na reálnych dátach. 1
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Cluster point processes in insurance mathematics
Veselá, Veronika ; Pawlas, Zbyněk (advisor) ; Dostál, Petr (referee)
Title: Cluster point processes in insurance mathematics Author: Veronika Veselá Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Abstract: In the present work we study point processes and their importance in insurance mathematics. With the help of cluster and marked point processes we can describe a model that considers times of claim occurence and times and hei- ghts of corresponding payments. We study two specific models which can be used to predict how much money is needed for claims which happened. The first model is chain ladder in the form of Mack's model. For this model we show chain ladder estimators of development factors, estimates of their variance and their proper- ties. We try to find one-step ahead prediction and multi-step ahead prediction, which we use for calculating prediction of reserves. We shortly review asymptotic properties of the estimators in Mack's model. The second model is the Poisson cluster model. Firstly we define this model and the variables entering the model. Then we devote attention to one-step ahead and multi-step ahead prediction. We also study prediction when some variables have specific distributions. Finally, we use both methods of prediction on simulated data and compare their average relative absolute errors....
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Random tessellations and their statistical analysis
Vook, Peter ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
Statistical aspects of random mosaics have not been heretofore given enough attention. This thesis deals with the derivation of estimators and statistical tests in a three-dimensional Poisson-Voronoi mosaic model. The first chapter compiles elementary results in the fields of point processes, random closed sets and particle processes. These are used in a second chapter to deduce geometric properties of random mosaics. The third chapter introduces the statistical research itself, estimators and model tests. Horvitz- Thompson estimator is introduced in order to correct statistics calculated on a reduced sample. Own results are tried in a computer simulation and compared to existing research in the last chapter. Mainly, the quality of estimators and the power of proposed tests is observed. 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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Random inscribed polygons
Kantor, Matěj ; Pawlas, Zbyněk (advisor) ; Nagy, Stanislav (referee)
In this work we study randomly inscribed polygons into the unit circle, par- ticularly the asymptotic properties of their area and perimeter. The results show that area, perimeter and their expected values can be used to approximate the number π. Further we present several approaches for improving the rate of conver- gence of these approximations. Some of which are based on constructing suitable 2n-sided random polygons together with combining different areas and perime- ters. Briefly we show a couple of results for a generalized d-dimensional case. Finally we verify the validity of the studied theoretical results on specific experi- ments implemented in the Matlab environment. 1
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Multivariate Extensions of Poisson Distribution
Růžička, Tomáš ; Komárek, Arnošt (advisor) ; Pawlas, Zbyněk (referee)
In this bachelor thesis we introduce several models of multivariate Poisson distribu- tion. At first we briefly mention univariate Poisson distribution and prove subsidiary theorem. We further introduce two models, which rely on properties of the univariate Poisson distribution. The second chapter is supplemented by proven theorems, which deal mainly with the calculation of probability. In the next chapter are proposed point estimators of the model parameters and derived their properties. Finally we demonstrate numerical calculation and three simulations. The last chapter summarizes some other models of multivariate Poisson distribution. 1
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