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# Uniform Random Number Generation

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## Abstract

In typical stochastic simulations, randomness is produced by generating a sequence of independent uniform variates (usually real-valued between 0 and 1, or integer-valued in some interval) and transforming them in an appropriate way. In this paper, we examine practical ways of generating (deterministic approximations to) such uniform variates on a computer. We compare them in terms of ease of implementation, efficiency, theoretical support, and statistical robustness. We look in particular at several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear congruential over fields of formal series, and combined generators, and show how all of them can be analyzed in terms of their lattice structure. We also mention other classes of generators, like non-linear generators, discuss other kinds of theoretical and empirical statistical tests, and give a bibliographic survey of recent papers on the subject.
... Example 15. In a box there are two coins, the first is fair and the second is biased so that p(H) = 1. ...
... Example 15. Let X and Y be two d. ...
... Definition 1. [15] The pseudo-random number generator (PRNG) is a structure (S, P 0 , f, U, g) such that: -S : a finite set of states (states space) s 0 , s 1 , · · · , s n -P 0 : probability distribution on S to select the initial state s 0 (seed) f : transition function f : S → S -U : space of generated numbers (output) (often [0, 1]) g : output function g : S → U The elements of this structure make it possible to perform the steps of random numbers generation, which are : 1-Select s 0 using P 0 , then generate the first random number u 0 = g(s 0 ) 2-At each step i ≥ 1, the transition function changes the state of the generator and then the output function gives the random number associated with the new state. ...
... The problem faced in the design of a noise generator system is how to generate a random signal that has a certain distribution (uniform or gaussian) efficiently. From the literature search, the authors get a lot of literature that writes about methods of generating random signals either purely (pure random) or artificial (pseudo random) [3]. ...
... The artificial noise generation method generally uses a mathematical algorithm (to generate a series of random numbers) which is then converted into an analog signal using a DAC (Digital to Analog Converter). The main weakness of the noise generation method using this mathematical algorithm is the appearance of periodization in the resulting signal [3][4] [5]. There is a lot of literature that focuses its research on developing algorithms to generate a series of random numbers close to the characteristics of pure random numbers. ...
... From Figure 1, it can be seen that to generate Gaussian distributed random numbers using the Box-Muller method, a uniformly distributed random number is required. To generate random numbers with uniform distribution, the Lehmer method [3][10] can be used. Mathematically, the Lehmer method for generating random numbers can be written as: ...
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The random noise signal is widely used as a test signal to identify a physical or biological system. In particular, the Gaussian distributed white noise signal (Gaussian White Noise) is popularly used to simulate environmental noise in telecommunications system testing, input noise in testing ADC (Analog to Digital Converter) devices as well as testing other digital systems. Random noise signal generation can be done using resistors or diodes. The weakness of the noise generator system using physical components is the statistical distribution. An alternative solution is to use a Pseudo-Random System that can be adjusted for distribution and other statistical parameters. In this study, the implementation of the Gaussian distributed pseudo noise generation algorithm based on the Enhanced Box-Muller method is described. Prototype of noise generation system using a minimum system board based on Cortex Microcontroller or MCU-STM32F4. From the test results, it was found that the Enhanced Box-Muller (E Box-Muller) method can be applied to the MCU-STM32F4 efficiently, producing signal noise with Gaussian distribution. The resulting noise signal has an amplitude of ±1Volt, is Gaussian distributed and has a relatively wide frequency spectrum. The noise signal can be used as a jamming device in a certain frequency band using an Analog modulator.
... Définition 1. [15] Le générateur des nombres pseudo-aleatoires (PRNG) est la structure (S, P 0 , f, U, g) tel que : -S : un ensemble fini d'états (Espace d'états) s 0 , s 1 , · · · , s n -P 0 : distribution de probabilité sur S pour séléctioner l'état initial s 0 (seed) f : fonction de transition f : S → S -U : espace des nombres générés (sortie) (souvent [0, 1]) g : fonction des sorties g : S → U Les éléments de cette structure permettent de réaliser les étapes de la génération des nombres aléatoires qui sont : 1-Sélectionner s 0 en utilisant P 0 , puis générer le premier nombre aléatoire u 0 = g(s 0 ) 2-À chaque étape i ≥ 1, la fonction de transition change l'état du générateur et ensuite la fonction des sorties donne le nombre aléatoire associé au nouvel état. ...
... Output ______________________________________ # Expected value: 3.9628993353781383 # Variance: 15.587130335771537 # Standard deviation: 3.9480539935228265 # Before normalization. ...
... Considering a vector of elements and the quicksort algorithm [53], sorting is of average time complexity log( ). Meanwhile, the random sequences used in the proposed algorithm are derived using a PRNG, the complexity cost of which can be computed by considering the well-known linear congruential generator (LCG) as a uniform random number generator [54]. The operations related to (15)-(18) represent the complexity of exchanging the sum of image pixels . ...
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