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Confidence intervals for parameters of multinomial distribution
Bárnetová, Kamila ; Anděl, Jiří (advisor) ; Omelka, Marek (referee)
Title: Confidence intervals for parameters of multinomial distribution Author: Kamila Bárnetová Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jiří Anděl, DrSc., Department of Probability and Mathematical Statistics Abstract: Confidence intervals for parameters for binomial and multinomial distribution are described in this thesis. These intervals can be used in practice, for exemple- pre-election estimates. The first two chapter are devoted to derivation of these intervals. Simulations and comparison of several selected methods can be found in the last chapter. Based on the simulations, we consider it appropriate, to choose to calculate confidence intervals for parameters of multinomial distribution intervals based on Bonferroniho inequality, or their modifications. These intervals are easy to calculate, while their coverage probability is at least 0.89. Keywords: confidence interval, multinomial distribution, binomial distribution, Bonferroni inequality
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Variance stabilizing transformations
Kuželová, Noemi ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
Abstract. We often examine data whose sample mean converges to a normal distribution, but the variance generally depends on an unknown parameter. To get rid of this dependence, we can sometimes use the so-called variance-stabilizing transformation method. Firstly, this thesis explains the method in detail and finds a general procedure to find suitable transformations. Then it will focus on data from Poisson and binomial distributions with unknown parameters. For these data, it finds transformations that stabilize (asymptotic) variance, and compares them with the "improved"transforms from the article Anscombe (1948). Most of the thesis is devoted to the shape of these transformations. Finally, we show in the Poisson distribution simulation that it is really appropriate to use this method and compare the derived transformation with its Anscombe version.
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Markov binomial model
Šuléřová, Natálie ; Hudecová, Šárka (advisor) ; Dvořák, Jiří (referee)
In this thesis we study the Markov chain binomial model, which generalizes the standard binomial distribution. Instead of the sum of independent random vari- ables, we consider the sum of random variables that form a stationary Markov chain. The goal of this thesis is to describe this model along with its proper- ties, such as the expected value, variance and probability generating function. A part of this thesis is dedicated to estimating parameters of this model using the method of moments and the maximum likelihood estimation. The accuracy of the methods is compared in a simulation study and obtained results are dis- cussed. The presented model is then applied on a real dataset based on rate of alcohol-impaired car accidents.
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Aplikace matematických znalostí při výuce biologie
STUDENÁ, Lucie
The Theses deals with applications of mathematical knowledge in teaching biology and it is divided into four chapters. Each chapter is dedicated to another application: 1. Application of conditional probability in medical diagnostics, 2. Application of exponential function in population ecology, 3. Application of logic functions in mathematical modelation of neuron and 4. Aplication of binomial theorem and binomial distribution in genetics. Each application contains solved problems, a worksheet for students and a solution for each worksheet. Two application (1. and 2.) have been tested in teaching and as an assessment of my lessons students filled questionnaires. Results of these questionnaires are processed in the end of these chapters. This Thesis can be used in teaching or self-studying.
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Confidence intervals for parameters of multinomial distribution
Bárnetová, Kamila ; Anděl, Jiří (advisor) ; Omelka, Marek (referee)
Title: Confidence intervals for parameters of multinomial distribution Author: Kamila Bárnetová Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jiří Anděl, DrSc., Department of Probability and Mathematical Statistics Abstract: Confidence intervals for parameters for binomial and multinomial distribution are described in this thesis. These intervals can be used in practice, for exemple- pre-election estimates. The first two chapter are devoted to derivation of these intervals. Simulations and comparison of several selected methods can be found in the last chapter. Based on the simulations, we consider it appropriate, to choose to calculate confidence intervals for parameters of multinomial distribution intervals based on Bonferroniho inequality, or their modifications. These intervals are easy to calculate, while their coverage probability is at least 0.89. Keywords: confidence interval, multinomial distribution, binomial distribution, Bonferroni inequality
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Interval estimates for binomial proportion
Borovský, Marko ; Zvára, Karel (advisor) ; Sečkárová, Vladimíra (referee)
The subject of this thesis is the point estimate and interval estimates of the binomial proportion. Interval estimation of the probability of success in a binomial distribution is one of the most basic and crucial problems in statistical practice. The thesis is divided into three chapters. The first chapter is about maximum- likelihood estimation for a binomial proportion. Futhermore, we will describe several methods of the construction of confidence intervals. In the end, we will compare all intervals in term of the actual coverage probability and expected length. 1
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Interval Estimate of Binomial Parameter p: What is (Relatively) New?
Klaschka, Jan
The present work follows up the ROBUST 2006 paper where various types of confidence intervals for binomial parameter p have been exposed. The coverage probability cannot equal the nominal confidence level 1-alpha in the whole domain [0, 1]. This leads to dilemmas (is the coverage of at least 1-alpha a must, or is it better to approximate 1-alpha from both sides?), and to multiplicity of proposals of confidence interval types. The present work extends the scope of the previous paper by such generalizations of "ordinary" confidence intervals that enable a constant coverage, namely by the randomized confidence intervals (introduced several decades ago), and by the relatively new idea of the fuzzy confidence intervals.
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O intervalových odhadech pravděpodobností, zvláště malých
Klaschka, Jan
Práce se zabývá modifikací Clopper-Pearsonova konfidenčního intervalu pro parametr binomického rozdělení při nulovém nebo plném počtu úspěchů. Ukazuje se, že modifikace, která má stále své přívržence, je nekorektní. Dále jsou připomenuty některé lepší alternativy k nejběžněji používaným typům konfidenčních intervalů.
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