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In this thesis our goal is to study pseudorandom numbers. We  will investigate how to produce pseudorandom samples from the uniform distribution with a method called the linear congruential method. Another method we will look at is the inverse sampling method which gives us the possibility to generate samples from other distributions that are not the uniform distribution. When generating pseudorandom samples quality is an important aspect, therefore we are going to take a look at a discrepancy which is a tool to determine quality of uniformly distributed samples. We implement the methods in Python and perform numerical experiments to test some quality aspects of the output.

Many physical, biological, ecological and behavioral events occur at times and rates that are exponentially distributed. Modeling these systems requires simulators that can accurately generate a large quantity of exponentially distributed random numbers, which is a computationally intensive task. To improve the performance of these simulators, one approach is to move portions of the computationally inefficient simulation tasks from software to custom hardware implemented in Field Programmable Gate Arrays (FPGAs). In this work, we study efficient FPGA implementations of exponentially distributed random number generators to improve simulator performance. Our approach is to generate uniformly distributed random numbers using standard techniques and scale them using the inverse cumulative distribution function (CDF). Scaling is implemented by curve fitting piecewise linear, quadratic, cubic, and higher order functions to solve for the inverse CDF. As the complexity of the scaling function increases (in terms of order and the number of pieces), number accuracy increases and additional FPGA resources (logic cells and block RAMs) are consumed. We analyze these tradeoffs and show how a designer with particular accuracy requirements and FPGA resource constraints can implement an accurate and efficient exponentially distributed random number generator.

In the given work the problem of quality of generating of random numbers is studied. The purpose of work - to find out opportunities and qualitative characteristics of various generators of random numbers; to test their work; to compare and estimate quality of most often used generators. Work consists of three basic parts: The first part is devoted to questions of generating of random numbers, namely: what is the random number where sequences of random numbers are used, what ways of their reception. The big attention is given a question of qualitative characteristics of generators of random numbers, their classification is resulted. The greatest attention is given program gauges pseudo random numbers, and the information on hardware devices of generating of casual sequences has fact-finding character. In the second part it is spoken about testing generators of random numbers. In this part the basic methods of testing are considered, the most interesting sets of statistical tests are described. The third part - research. The purpose of researches - to allocate from the most popular program generators of random numbers (standard functions of various programming languages: Basic, Pascal, Delphi, C ++), the generator with as much as possible high quality of generating. In the end of work conclusions about the executed researches are given. Texts of the used programs, full the table of data, schedules and diagrams are presented in appendices.

Many real-world applications use random numbers generated by pseudo-random number and true random number generators (TRNG). Unlike pseudo-random number generators which rely on an input seed to generate random numbers, a TRNG relies on a non-deterministic source to generate aperiodic random numbers. In this research, we develop a novel and generic software-based TRNG using a random source extracted from compute architectures of today. We show that the non-deterministic events such as race conditions between compute threads follow a near Gamma distribution, independent of the architecture, multi-cores or co-processors. Our design improves the distribution towards a uniform distribution ensuring the stationarity of the sequence of random variables. We improve the random numbers statistical deficiencies by using a post-processing stage based on a heuristic evolutionary algorithm. Our post-processing algorithm is composed of two phases: (i) Histogram Specification and (ii) Stationarity Enforcement. We propose two techniques for histogram equalization, Exact Histogram Equalization (EHE) and Adaptive EHE (AEHE) that maps the random numbers distribution to a user-specified distribution. EHE is an offline algorithm with O(NlogN). AEHE is an online algorithm that improves performance using a sliding window and achieves O(N). Both algorithms ensure a normalized entropy of (0:95; 1:0]. The stationarity enforcement phase uses genetic algorithms to mitigate the statistical deficiencies from the output of histogram equalization by permuting the random numbers until wide-sense stationarity is achieved. By measuring the power spectral density standard deviation, we ensure that the quality of the numbers generated from the genetic algorithms are within the specified level of error defined by the user. We develop two algorithms, a naive algorithm with an expected exponential complexity of E[O(eN)], and an accelerated FFT-based algorithm with an expected quadratic complexity of E[O(N2)]. The accelerated FFT-based algorithm exploits the parallelism found in genetic algorithms on a homogeneous multi-core cluster. We evaluate the effects of its scalability and data size on a standardized battery of tests, TestU01, finding the tuning parameters to ensure wide-sense stationarity on long runs. / October 2016

The aim of the Thesis is to determine the influence of technical analysis to the profitability of shares by O2 Telefónica, Komerční banka and ČEZ whan applied to a predeterminated trading model and judge the validity of Efective Markets Theory. It will be also followed up the influence of trading signals of technical indicators on the shape and characteristics of conditional distribution set by those signals. The Crystal Ball was used as the main software tool. This software dispose of all tools necesarry for the analysis. There is also desribed the theoretical background of areas like shares, technical analysis, random quantity distribution, generating of random numbers and other related areas in the Thesis, so the reader who does not understand the problemacy could understand the best.

Derivativos exóticos são produtos com estrutura complexa e personalizada cujo apreçamento pode requerer o uso de simulações de Monte Carlo. Todavia, essas simulações têm alto custo computacional, o que torna lento o apreçamento de uma carteira com vários derivativos. Para mitigar esse problema, propõe-se o reuso de números aleatórios entre diferentes operações de uma mesma carteira apreçada através do método de Monte Carlo. Realiza-se o apreçamento de cinco carteiras de derivativos exóticos com duas implementações da simulação de Monte Carlo, uma sem e outra com reuso de números aleatórios. Observa-se que, quanto mais operações há na carteira, maior é a vantagem de performance da estratégia com reuso em relação à outra abordagem de implementação. O erro quadrático médio do preço dos derivativos obtidos através das simulações em relação ao preço teórico esperado mantém-se o mesmo em ambas as implementações. Portanto, é possível sugerir que o algoritmo com reuso de número aleatórios apresenta uma maneira de melhorar a performance do método de Monte Carlo sem aumentar o erro da simulação. / Exotic derivatives are products with complex and customized structure whose pricing may require the use of Monte Carlo simulation. However, this kind of simulation has high computational cost, which slows the pricing of a portfolio containing several derivatives. In order to mitigate this problem, it is proposed the reuse of random numbers across different trades in the same portfolio priced using the Monte Carlo method. Five portfolios of exotic derivatives are priced using two implementations of Monte Carlo simulation, with and without reuse of random numbers. It is observed that the more trades are in the portfolio, the better is the performance of the reuse approach compared to the regular implementation. The mean squared error of simulation prices compared to the theoretical value remain the same in both implementations. Therefore, it is possible to suggest that the algorithm which reuses random numbers presents a way to improve Monte Carlo method performance with no increment of simulation error.

Derivativos exóticos são produtos com estrutura complexa e personalizada cujo apreçamento pode requerer o uso de simulações de Monte Carlo. Todavia, essas simulações têm alto custo computacional, o que torna lento o apreçamento de uma carteira com vários derivativos. Para mitigar esse problema, propõe-se o reuso de números aleatórios entre diferentes operações de uma mesma carteira apreçada através do método de Monte Carlo. Realiza-se o apreçamento de cinco carteiras de derivativos exóticos com duas implementações da simulação de Monte Carlo, uma sem e outra com reuso de números aleatórios. Observa-se que, quanto mais operações há na carteira, maior é a vantagem de performance da estratégia com reuso em relação à outra abordagem de implementação. O erro quadrático médio do preço dos derivativos obtidos através das simulações em relação ao preço teórico esperado mantém-se o mesmo em ambas as implementações. Portanto, é possível sugerir que o algoritmo com reuso de número aleatórios apresenta uma maneira de melhorar a performance do método de Monte Carlo sem aumentar o erro da simulação. / Exotic derivatives are products with complex and customized structure whose pricing may require the use of Monte Carlo simulation. However, this kind of simulation has high computational cost, which slows the pricing of a portfolio containing several derivatives. In order to mitigate this problem, it is proposed the reuse of random numbers across different trades in the same portfolio priced using the Monte Carlo method. Five portfolios of exotic derivatives are priced using two implementations of Monte Carlo simulation, with and without reuse of random numbers. It is observed that the more trades are in the portfolio, the better is the performance of the reuse approach compared to the regular implementation. The mean squared error of simulation prices compared to the theoretical value remain the same in both implementations. Therefore, it is possible to suggest that the algorithm which reuses random numbers presents a way to improve Monte Carlo method performance with no increment of simulation error.

The transformed rejection method, a combination of inversion and rejection, which can be applied to various continuous distributions, is well suited to generate binomial random variates as well. The resulting algorithms are simple and fast, and need only a short set-up. Among the many possible variants two algorithms are described and tested: BTRS a short but nevertheless fast rejection algorithm and BTRD which is more complicated as the idea of decomposition is utilized. For BTRD the average number of uniforms required to return one binomial deviate lies between 2.5 and 1.4 which is considerably lower than for any of the known uniformly fast algorithms. Timings for a C-implementation show that for the case that the parameters of the binomial distribution vary from call to call BTRD is faster than the current state of the art algorithms. Depending on the computer, the speed of the uniform generator used and the binomial parameters the savings are between 5 and 40 percent. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

This thesis is focused on exploring the sources of entropy. It includes a description of random number generators and tests used to evaluate entropy quality. Random number generator for Windows and Linux OS was created together with software for entropy evaluation. Subsequently, measurement of entropy was performed on physical workstations and Cloud environments.