|
|
The beginnings of probability theory
Marcinčín, Martin ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
The purpose of this thesis is to give a summary of historical development and explain fundamentals of the probability theory. Early systematic thoughts, emergence of classical Laplace, geometric and statistical definition of probability with development of theory, independence, conditional probability and Bayes theorem are shown. The thesis describes first mention of random values and the central limit theorem. The alternative, discrete uniform, binomial, Poisson, continuous uniform, normal and exponential distributions are discussed with historical background of their discoveries. The theory is supplemented with illustrative and contemporary examples. The thesis describes development in various fields of probability until publication of the Kolmogorov's probability theory in 1933. Powered by TCPDF (www.tcpdf.org)
|
|
|
Buffon needle problem and its generalizations
Hledík, Jakub ; Pawlas, Zbyněk (advisor) ; Prokešová, Michaela (referee)
This thesis contains detailed derivation of results of several generalizations of the Buffon needle problem. Next to the original problem we study grids composed of rectangles, known as Buffon-Laplace needle problem, then grids composed of parallelograms, triangles or hexagons. The application of this problem is briefly shown on the estimation of π, additional references are mentioned. We provide a proof of the theorem computing the area of a polygon, if the Cartesian coordi- nates of its vertices are known. Finally, we show how to solve grids composed of several different shapes, this is demonstrated on the grid composed of a regular hexagon and a diamond. 1
|
|
|
Covering the circle by random arcs
Čelikovská, Klára ; Pawlas, Zbyněk (advisor) ; Dvořák, Jiří (referee)
In this thesis we consider the geometric probability problem of covering a circle with random arcs. We randomly place arcs of a fixed length on a circle of unit circumference. First we find the probability of covering the entire circle with a finite number of arcs of the same length and show some of its numerical values. Next we study the random variable describing the size of the covered part of the circle and the expected number of arcs needed to fully cover the circle if we place the arcs sequentially. Finally, we solve a similar problem of covering the circle by a countably infinite number of arcs of different lengths. 1
|
|
|
The beginnings of probability theory
Marcinčín, Martin ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
The purpose of this thesis is to give a summary of historical development and explain fundamentals of the probability theory. Early systematic thoughts, emergence of classical Laplace, geometric and statistical definition of probability with development of theory, independence, conditional probability and Bayes theorem are shown. The thesis describes first mention of random values and the central limit theorem. The alternative, discrete uniform, binomial, Poisson, continuous uniform, normal and exponential distributions are discussed with historical background of their discoveries. The theory is supplemented with illustrative and contemporary examples. The thesis describes development in various fields of probability until publication of the Kolmogorov's probability theory in 1933. Powered by TCPDF (www.tcpdf.org)
|
|
|
Geometric probability
Březinová, Eliška ; Malá, Ivana (advisor) ; Čabla, Adam (referee)
This thesis deals with geometric probability applied on practical exercises. It covers Buffon's needle problem in detail; Laplace's conclusions about pi are supported by my own trial. Next, Bertrand's paradox is solved, and the conclusions are demonstrated on computer programs, which simulate the experiment. One chapter is dedicated to eight different exercises, which can be often found in textbooks. In the end we will mention practical usage of geometric probability, especially in the medicine field. We will point out to usage of modified Buffon's principle, which is used to estimate lengths of planar structures.
|
|
| |
guest ::
