Search

In this paper, we mainly investigate the random convolution sampling stability for signals in multiply generated shift invariant subspace of weighted mixed Lebesgue space. Under some restricted conditions for the generators and the convolution function, we conclude that the defined multiply generated shift invariant subspace could be approximated by a finite dimensional subspace. Furthermore, with overwhelming probability, the random convolution sampling stability holds for signals in some subset of the defined multiply generated shift invariant subspace when the sampling size is large enough.

Aiming at the existing problem of losing spatial details and spectral information in remote sensing image fusion, a method of remote sensing image fusion based on convolution sampling transformation is proposed. Firstly, two images to be fused are convoluted, sampled, filtered hierarchically, and decomposed into different sub-images at different levels. Next, those sub-images are fused according to the corresponding locations, and then reconstructed to be a fused image. For images with more complex terrains and objects, the requirements for spatial details and spectral information are higher. So we propose to further transform the fused image and panchromatic image in space. The first component of the former is replaced by that of the latter. Finally, the fused image is obtained after performing inverse transformation. The experimental results show that the fusion effect of two proposed methods is better than that of traditional image fusion algorithms, such as PCA transform, HIS transform, wavelet transform and so on. Compared with the previous method, the latter method has higher resolution but less spectral information in the case of more complex terrains and objects for two proposed methods. They are effective remote sensing image fusion methods.

Cryptosystem recognition is an important link in cryptanalysis, and ciphertext feature extraction is very important in cryptosystem recognition. In this paper, a ciphertext feature extraction method based on linear transformation and convolution sampling is proposed. A large number of plaintext data are encrypted as the data set, and a random forest is used to construct a classifier to recognize the two-class classification and multi-class classification of four cryptographic systems, such as AES, 3DES, Blowfish and RSA. The experimental results show that the recognition performance of convolution features is better and the accuracy of recognition in multi-class classification scene is improved. The accuracy of two class classification recognition is more than 80% and that of multi-class classification recognition is more than 70%.

The combination of single-pixel-imaging and single-photon-counting technology can achieve ultrahigh-sensitivity photon-counting imaging. However, its applications in high-resolution and real-time scenarios are limited by the long sampling and reconstruction time. Deep-learning-based compressive sensing provides an effective solution due to its ability to achieve fast and high-quality reconstruction. This paper proposes a sampling and reconstruction integrated neural network for single-photon-counting compressive imaging. To effectively remove the blocking artefact, a subpixel convolutional layer is jointly trained with a deep reconstruction network to imitate compressed sampling. By modifying the forward and backward propagation of the network, the first layer is trained into a binary matrix, which can be applied to the imaging system. An improved deep-reconstruction network based on the traditional Inception network is proposed, and the experimental results show that its reconstruction quality is better than existing deep-learning-based compressive sensing reconstruction algorithms.

An oscillator-based true random number generator (TRNG) is experimentally demonstrated by exploiting inherently stochastic threshold switching in the insulator-to-metal transition (IMT) in vanadium dioxide. Through experimentation and modeling, we show that the origin of stochasticity arises from small perturbations in the nanoscale domain structure, which are then subsequently amplified through a positive feedback process. Within a 1T1R oscillator, the stochastic cycle-to-cycle variations in the IMT trigger voltage result in random timing jitter, which is harnessed for a TRNG. The randomness of the IMT TRNG output is validated using the NIST SP800-22 statistical test.

We derive novel exact closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the product and ratio of products of an arbitrary number of independent non-identically distributed (i.n.i.d) extended generalized- $$\mathcal {K}$$ (EGK) variates. The expressions are given in terms of the Meijer’s G-function and can be computed easily using commonly available mathematical software tools. They also subsume those for arbitrary combinations of other well-known variates and can be directly utilized in performance evaluation of wireless communication systems under different scenarios. We present various analytical results that are verified via Monte-Carlo simulations for both the PDF and CDF as well as their application in multiple practical scenarios.

We present a computationally efficient method to generate random variables from a univariate conditional probability density function (PDF) derived from a multivariate α-sub-Gaussian (αSG) distribution. The approach may be used to sequentially generate variates for sliding-window models that constrain immediately adjacent samples to be αSG random vectors. We initially derive and establish various properties of the conditional PDF and show it to be equivalent to a Student's t-distribution in an asymptotic sense. As the αSG PDF does not exist in closed form, we use these insights to develop a method based on the rejection sampling (accept-reject) algorithm that allows generating random variates with computational ease. HighlightsAn efficient method to generate random variates for a-sub-Gaussian processes with memory is presented.Properties of the univariate conditional α-sub-Gaussian distribution are investigated.Convergence of the aforementioned distribution to a Student's t-distribution is proven in an asymptotic sense.Using the above properties and tabulation of a heavy-tailed function, rejection sampling is used to generate realizations.The method may be used in simulation-based performance analysis of systems operating in colored α-sub-Gaussian noise.

In general, the exact distribution of a convolution of independent gamma random variables is quite complicated and does not admit a closed form. Of all the distributions proposed, the gamma...

In this article, a convolution of second-order random measures on a locally compact abelian groups is considered. A random measure is an analog of a stochastic process. With a newly defined (random) convolution, we construct a ring and a normed algebra of second-order random measures.

This paper presents a new approach to analysis and design of ADC-based random number generators. To this end, different full-bit and half-bit redundant stages of algorithmic converter are used to design chaotic maps. It is shown that, in the redundant and nonredundant structures, output probability density function of the converter stages and their related chaotic functions always converge to uniformity. It is demonstrated that residues become independent and uniformly distributed. This fact leads to the randomness and uniformity of distribution of the random number generator output bits. Moreover, it is shown that some common chaotic maps that are employed in chaotic random number generators can be implemented using nonredundant and half-bit redundant stages of algorithmic converter. In this way, the capability of ADC-based generators in designing chaotic maps and producing random number sequences is illustrated. The validity of the proposed chaos-based random number generator is confirmed using NIST statistical tests even in the presence of nonidealities in algorithmic converter. Since the ADCs are mixed-signal integrated circuits and can be used in high-speed applications, the ADC-based random number generator has high throughput and is easily embeddable in all analog and digital circuits.

A four-parameter probability distribution, which includes a wide variety of curve shapes, is presented. Because of the flexibility, generality, and simplicity of the distribution, it is useful in the representation of data when the underlying model is unknown. Further important applications of the distribution include the modeling and subsequent generation of random variates for simulation studies and Monte Carlo sampling studies of the robustness of statistical procedures. This research centered on combining these two distributions that will simultaneously capture the rate of occurrence of a phenomenon, especially buying behaviour and the actual performance of that phenomenon as well as tracking and forecasting future purchasing pattern based the data. Further important applications of the distribution include the modeling and subsequent generation of random variates for simulation studies of the robustness of statistical procedures. To do this, specification of the hybrid model named Exponential- Gamma mixture model is given and followed by its derivation. The concluding part of the paper depicts an example of the areas of its application.

A new ring-oscillator-based true random number generator (RNG) based on multiple sampling is proposed. The proposed generator uses the outputs of all gates in a ring oscillator, as positive- and negative-edge clock signals. The generator in two field-programmable gate array families is implemented, and it is verified that the generated bit sequences pass the RNG tests of the National Institute of Standards and Technology. The experimental results show that the proposed generator is faster and more compact than the existing ring-oscillator-based true RNGs.

In this paper, the convolution and the order statistics of k independent random lifetimes are considered as random times. Based on the concept of residual life at such random times, new characterizations of the exponential distribution are established.

Assume that xi(1), xi(2),... are independent and identically distributed non-negative random variables having the O-exponential distribution. Suppose that eta is a nonnegative non-degenerate at zero integer-valued random variable independent of xi(1), xi(2),.... In this paper, we consider the conditions for eta under which the distribution of random sum xi(1) + xi(2) + ... + xi(eta) remains in the class of O-exponential distributions.

International audience; We propose a matrix analysis approach to analytically provide the cumulative distribution function of the sum of independent Erlang random variables. This reduces to the characterization of the exponential of the involved generator matrix. We propose a particular basis of vectors in which we write the generator matrix. We find, in the new basis, a Jordan–Chevalley decomposition allowing to simplify the calculation of the exponential of the generator matrix. This is a simpler alternative approach to the existing ones in the literature.

Adaptive Rejection Sampling (ARS) method is a well-known random number generator to acquire a random sample from a probability distribution, and has the advantage of improving the proposal distribution during the sampling procedures, which update it closer to the target distribution. However, the use of ARS is limited since it can be used only for the target distribution in the form of the log-concave function, and thus various methods have been proposed to overcome such a limitation of ARS. In this paper, we attempt to compare five random number generators based on ARS in terms of adequacy and efficiency. Based on empirical analysis using simulations, we discuss their results and make a comparison of five ARS-based methods.

This paper is devoted to a pseudo-random number generator that generates numbers according to a known probability density function. This function has its origins in a histogram that contains the observed properties of a monitored random variable X. Using the recorded PDF and an ON/OFF model we are able to generate output states of a random variable Y, which is nearly alike distributed as the random variable X. Because the used PDF has its origins in a normed histogram function, the resulting number of all possible output states of variable Y is the same as the number of classes that are used to form the histogram. We propose this model that generates practically an unlimited set of possible output states of the random variable Y. At first a cumulative distribution function is formed from the input density function. Then the ON/OFF model driven by a random number generator in (0, 1) is applied to address classes that are defined in the distribution function. To address these classes a search function is applied. Then when a class is chosen, we propose to use a second uniform random number generator that generates numbers within the range of the selected class. The resulting output state of the variable Y is the output of the second random number generator. In this paper we apply this model on a packet generator that generates IPTV inter-departure times of an MPEG2-TS H.264 VBR based video source.

An algorithm for generating random variates from the Madland–Nix fission energy spectrum assuming a constant compound nucleus cross section is given based on physics considerations. A program was written to generate variates using the algorithm developed, and it was shown that the generated variates match the probability density function. This algorithm can be used by Monte Carlo particle transport codes to sample secondary energies for neutrons born from fission when the underlying data is given as parameters to a Madland–Nix energy spectrum.

In compressed sensing and sparse representation-based pattern recognition, random projection with a dense random transform matrix is widely used for information extraction. However, the complicated structure makes dense random matrices computationally expensive and difficult in hardware implementation. This paper considers the simplification of the random projection method. First, we propose a simple random method, random combination, for information extraction to address the issues of dense random methods. The theoretical analysis and the experimental results show that it can provide comparable performance to those of dense random methods. Second, we analyze another simple random method, random choosing, and give its applicable occasions. The comparative analysis and the experimental results show that it works well in dense cases but worse in sparse cases. Third, we propose a practical method for measuring the effectiveness of the feature transform matrix in sparse representation-based pattern recognition. A matrix satisfying the Representation Residual Restricted Isometry Property can provide good recognition results.

Phase-type (PH) distributions are being used to model a wide range of phenomena in performance and dependability evaluation. The resulting models may be employed in analytical as well as in simulation-driven approaches. Simulations require the efficient generation of random variates from PH distributions. PH distributions have different representations and different associated computational costs for pseudo random-variate generation (PRVG). In this paper we study the problem of efficient representation and efficient generation of PH distributed variates. We introduce various PH representations of different sizes and optimize them according to different cost functions associated with PRVG.

The accurate and efficient simulation of random heterogeneous media is important in the framework of modeling and design of complex materials across multiple length scales. It is usually assumed that the morphology of a random microstructure can be described as a non-Gaussian random field that is completely defined by its multivariate distribution. A particular kind of non-Gaussian random fields with great practical importance is that of translation fields resulting from a simple memory-less transformation of an underlying Gaussian field with known second-order statistics. This paper provides a critical examination of existing random field models of heterogeneous two-phase media with emphasis on level-cut random fields which are a special case of translation fields. The case of random level sets, often used to represent the geometry of physical systems, is also examined. Two numerical examples are provided to illustrate the basic features of the different approaches.

Real random values have wide range of application in different field of computer science such as cryptography, network security and communication, computer simulation, statistical sampling, etc. In purpose of generating real random values, need for a natural noisy source refers to the main challenge where a source of noise may be reliable for using in random number generator if and only if be derived from physical environment. In this work, we address this requirement by using DNA computing concepts where the molecular motion behavior of DNA molecular provides a pure source of physical noise that may be used for generating high quality real random values. Since one of the main factor for evaluating quality of real random values refer to expectation for generating approximately same amount of 0s and 1s, in this article we model a DNA-based random number generator in sticker mode with ability of generating equal numbers of 0 and 1. After using molecular motion behavior of DNA molecular as the natural source of noise into the proposed DNA-based random number generator, the generated value were subjected to frequency, run, and serial tests which are proposed by National Institute of Standards and Technology (NIST) for randomness evaluation. Obtained result from this evaluation shows that beside the achieving high scores in run and serial tests, the values generated by our DNA-based random number generator pass frequency test with 100% success.

In conventional cumulant method of probabilistic load flow (PLF), random generator outages are usually simulated by discrete distributions of nodal power injections, but that will lead to significant error in AC load flow model. An improved PLF method base on AC model is proposed in this paper, which considers random generator outages and loads uncertainties. Cumulant and Gram-Charlier series expansion were applied to deal with the random variations of loads, instead of convolution calculations. According to the characteristics and focused aspects of power grid, certain generators were selected to form event group of generator outages and each event was analysed by exact load flow.Then total probability theorem was introduced to obtain the probabilistic distributions of node voltages and line flows that considered random factors of loads and generators.The case study of IEEE 39-bus system shows that the random generator outages remarkably affect the probabilistic distributions of state variables. The proposed method can avoid the error caused by generator outages in conventional cumulant method. Furthermore, the result of proposed method is consistent with that of Monte Carlo simulation, while computation speed is much faster.

Linear Feedback Shift Register based Unique Random Number Generator is an enhancement of Random Number generator with the additional property that any number generated by a unique random number generator can’t be duplicated. As per users demand for not duplicated random numbers in some applications like transferring a random number over the network on the behalf of actual character of the message for security point of view, existence of unique random number generators are very essential. In this paper LFSR [1] (Linear Feedback Shift Register) is used to implement the proposed concept of unique random number generator. Using LFSR is a faster approach for generating random sequences because it requires only X-OR operations and shift registers that’s why its implementation is very easy in software as well as in hardware [2, 3]. We can easily modify the LFSR and produce different random sequences. So it is the best option for using LFSR in unique random number generator.

A software package, rBeta2009, developed to generate beta random numbers and Dirichlet random vectors in R is presented. The package incorporates state-of-the-art algorithms so as to minimize the computer generation time. In addition, it is designed in a way that (i) the generation efficiency is robust to changes of computer architecture; (ii) memory allocation is flexible; and (iii) the exported objects can be easily integrated with other software. The usage of this package is then illustrated and evaluated in terms of various performance metrics.

Constrained random simulation is becoming the mainstream methodology to verify system-wide properties in functional verification. It is a must to develop a high-throughput constrained random pattern generator, which is able to support arbitrary distribution. In this paper, we propose a novel range-splitting heuristic and a solution-density estimation technique to conquer the challenges of random pattern generators proposed in the recent literature. The solution densities can significantly increase by pruning infeasible subspaces. On the other hand, the estimated solution densities stored on a range-splitting tree statistically predict the distribution of solutions. Therefore, the generated patterns are ensured to meet the desired distribution with high throughput. Experimental results show that our framework achieves more than 10X speedup on average when compared to a commercial generator.

The purpose of this work is to speed up simulations of neural tissues based on the stochastic version of the Hodgkin–Huxley model. Authors achieve that by introducing the system providing random values with desired distribution in simulation process. System consists of two parts. The first one is a high entropy fast parallel random number generator consisting of a hardware true random number generator and graphics processing unit implementation of pseudorandom generation algorithm. The second part of the system is Gaussian distribution approximation algorithm based on a set of generators of uniform distribution. Authors present hardware implementation details of the system, test results of the mentioned parts separately and of the whole system in neural cell simulation task.

We present novel general, simple, exact, and closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the ratio of the products of two independentα-μvariates, where all variates have identical values of alpha parameter. Obtained results are applied in analysis of multihop wireless communication systems in different fading transmission environments. The proposed theoretical analysis is also complemented by various graphically presented numerical results.

A random number generator produces a periodic sequence of numbers on a computer. The starting point can be random, but after it is chosen, everything else is deterministic. A random number generator produces a periodic sequence of numbers on a computer. The starting point can be random, but after it is chosen, everything else is deterministic. This paper presents the entropy and p-value tests performed on classical and quantum random number generators, in order to check the randomness of the generated output data. Both generators have been integrated in software applications reproducing asymmetrical cryptosystems, their results contributing to the generation of key material within Diffie-Hellman protocol. Can regard these tests as a first step in determining if a generator is suitable for a particular cryptographic application. At the same time, the statistical tests cannot serve as a substitute for cryptanalysis.

International audience; Generating random numbers is essential in manycryptographic applications like key generation, cryptographicprotocols for example Transport Layer Security (TLS) protocol,nonce and also in Internet for choosing TCP sequence numbers.We need generators which are able to construct large amountsof secure random numbers. To this end, True Random NumberGenerators (TRNGs) which extract randomness from physicalprocesses are usually used. The sequences generated by TRNGscannot be reproduced. However, generating random numbersby this way is time-consuming and expensive. Another way togenerate random numbers is to use deterministic random numbergenerators in which the seed is reseeded many times during thegeneration of the sequence. In this paper, we propose a newpseudo-chaotic number generator (PCNG) that produces randomnumbers. The algorithm is refreshed many times by using entropysource from Linux kernel. Results of statistical properties (i.e.Nist test, auto and cross-correlation, histogram, chi2-test andsoftware security analysis) exhibit good performance thus demonstratingthat the proposed generator can be used confidently toproduce random numbers.

This chapter begins by introducing the concepts of pseudo-random and quasi-random sequences. The convergence rates for Monte Carlo integration using both uniformly distributed quasi-random and pseudo-random numbers are presented. Techniques for creating independent random variates with Normal, Lognormal and Student’s t t -distributions are then discussed. The generation of correlated multivariate random numbers is then described using either the Cholesky factorization of the correlation matrix or its eigen decomposition. There is then a description of how to repair a correlation matrix that is not positive definite, using either its eigen decomposition or a method based on optimized hyperspherical points. This chapter ends with a discussion of how to generate multivariate normal and lognormal variates which have a specfied mean and covariance matrix.

We consider convolution sampling and reconstruction of signals in certain reproducing kernel subspaces of L p , 1 ≤ p ≤ ∞ L^p, 1\le p\le \infty . We show that signals in those subspaces could be stably reconstructed from their convolution samples taken on a relatively separated set with small gap. Exponential convergence and error estimates are established for the iterative approximation-projection reconstruction algorithm.

This paper pointed out that we can determine the initialization x0 when the random sequence is generated by the piecewise linear chaotic map in finite precision, and aimed at solving this objection, an improved measure of perturbing the system variable timely is proposed, and an improved random number generator based on chaotic map is designed and realized. Meanwhile, their working process and theory are analyzed, and the simulation, validation and test of the whole system is finished, and the result indicated that the bit rates is up to 20 MHz and the random number could pass the FIPS140-2 randomness test.

We propose a foundational model to explain properties of the retention time distribution of particle transport in a random medium. These particles are captured and released by distributed theoretical plates in a random medium as in standard chromatography. Our approach differs from current models, since it is not based on simple random walks, but on a directed and coordinated movement of the particles whose retention time dispersion in the column is due to the imprisonment time of the particle spent in the theoretical plates. Given a pair of fundamental parameters ( λ c , λ e ) the capture and release probabilities, we use simple combinatorial methods to predict the Probability Distribution of the retention times. We have analyzed several distributions typically used in chromatographic peak fits. We show that a log–normal distribution with only two parameters describes with high accuracy chromatographic distributions typically used in experiments. This distribution show a better fit than distributions with a larger number of parameters, possibly allowing for better control of experimental data.

In this article, we show that combining random generators by group operation improves the statistical properties of the composite. It gives an effective way of finding random generators more and more close to the uniform. Moreover, we obtain an effective estimation of the speed of convergence to the uniform generator.

True random number generators, which output truly random numbers by extracting entropy from physical phenomena such as thermal and electronic noises, are widely used in the field of the cryptographic communication systems. We have been developing a superconductive true random number generator that can generate truly random number sequences, impossible to be predicted, by utilizing the high-speed operation and high-sensitivity of superconductive integrated circuits. In this study, we have calculated the dependences of correlation of output random bits on the generation rate. Statistical tests have been performed on the basis of the NIST statistical test suite in order to evaluate the quality of the randomness of sequences generated by the superconductive true random number generator at high generation rate. We have generated a random number sequence consisting of 3.2 M-bit at the generation rate of 30 Gbit/s using the superconductive true random number generator, fabricated by the ISTEC-SRL 2.5 kA/cm2 Nb standard process. The generated random number sequences passed 13 kinds of the statistical tests in the NIST statistical test suit, although the 3 tests were not performed because of the shortage of the generated random numbers. The result sufficiently proves that a superconductive true random number generator can generate a high quality of random numbers that can be used for practical cryptographic applications, at a generation rate of up to 30 Gbit/s.

In this paper, a random fuzzy shock model and a random fuzzy fatal shock model are proposed. Then bivariate random fuzzy exponential distribution is derived from the random fuzzy fatal shock model. Furthermore, some properties of the bivariate random fuzzy exponential distribution are proposed. Finally, an example is given to show the application of the bivariate random fuzzy exponential distribution.

The inversion method for generating non-uniformly distributed random variates is a crucial part in many applications of Monte Carlo techniques, e.g., when low discrepancy sequences or copula based models are used. Unfortunately, closed form expressions of quantile functions of important distributions are often not available. The (generalized) inverse Gaussian distribution is a prominent example. It is shown that algorithms that are based on polynomial approximation are well suited for this distribution. Their precision is close to machine precision and they are much faster than root finding methods like the bisection method that has been recently proposed.

Random number is widely used in computer application. What is used commonly in practice is pseudo-random number generated by mathematical algorithm. Several useful pseudo-random number generating algorithm and random testing methods are discussed. And according to the disadvantage of traditional algorithm, improvement of pseudo-random number generating algorithm and some new methods to generate real-random number are summarized and analyzed.

An estimation of the rate of convergence of the distributions of asymptotically normal statistics based on a random sample of random size to the Laplace distribution is obtained. It is assumed that the random size of a sample does not depend on the members of the sample and has a number of special properties.

In this article, we describe a new yet simple statistical procedure to better assess the quality of pseudo-random number generators. The new procedure builds on the statistical test suite proposed by the National Institute of Standards and Technology (NIST) and is especially useful for applications in economics. Making use of properties of the binomial distribution, we estimate the conjoint significance level of the test. We apply the proposed procedure to several well-known pseudo-random number generators, and the results confirm its effectiveness.

Some classes of applied problems from random field theory are discussed and methods to solve them are proposed. Examples from various fields of science and technology are considered.

The requirement of physical true random number generator and the difference between the true random number and pseudo-random number are presented. The recent experiements of true ran-dom number generator, expetically, the various schemes of true random number generator based on lasers, are summarized. Also, we report the progress of the thoretical and experimental research on physical true random number generator in our group.

The inverse Gaussian distribution is a useful distribution with important applications. But there is less discussion in the literature on sampling of this distribution. The method given in [Atkinson, A.C., 1982. The simulation of generalized inverse Gaussian and hyperbolic random variables. SIAM Journal on Scientific and Statistical Computing 3(4), 502-515] is based on rejection method where some (uniform) random numbers from the sample are discarded. This feature makes it difficult to take advantage of the low discrepancy sequences which have important applications. In [Michael, J., Schucany, W., Haas, R., 1976. Generating random variates using transformations with multiple roots. The American Statistician 30(2), 88-90], Michael et al. give a method to generate random variables with inverse Gaussian distribution. In their method, two pseudorandom numbers uniformly distributed on (0, 1) are needed in order to generate one inverse Gaussian random variable. In this short paper, we present a new method, based on direct approximate inversion, to generate the inverse Gaussian random variables. In this method, only one pseudorandom number is needed in generating one inverse Gaussian random variate. This method enables us to make use of the better convergence of low discrepancy sequence than the pseudorandom sequence. Numerical results show the superiority of low discrepancy sequence than the pseudorandom sequence in simulating the mean of the inverse Gaussian distribution by using our sampling method. Further application of this method in exotic option pricing under the normal inverse Gaussian model is under investigation.

True random number generators are in general more secure than pseudo random number generators. In this paper, we propose a novel true random number generator which generates a 256-bit random number by computer mouse movement. It is cheap, convenient and universal for personal computers. To eliminate the effect of similar movement patterns generated by the same user, three chaos-based approaches, namely, discretized 2D chaotic map permutation, spatiotemporal chaos and “MASK” algorithm, are adopted to post-process the captured mouse movements. Random bits generated by three users are tested using NIST statistical tests. Both the spatiotemporal chaos approach and the “MASK” algorithm pass the tests successfully. However, the latter has a better performance in terms of efficiency and effectiveness and so is more practical for common personal computer applications.

It is shown that any uniform fractional order statistic can be represented in distribution by a random convex combination of any number of neighboring uniform fractional order statistics with weights being products of independent beta distributed random variables. In particular, the result is valid for ordinary order statistics.

The way engineering is being taught has changed in recent years with the introduction of commercial and educational software thus allowing a better understanding of the subject matter and increased teaching efficiency. This paper presents the educational software package designed to work in PCs and provides an attractive way to introduce the fundamentals of random variables. The software can be used as a complement to theoretical classes or alone as a self-study tool. The package was implemented in MATLAB and uses MATLAB™ tool "makeshow" which allows the students to create interactive slideshows without building their own graphic interface. The software is easy to use and is in accordance with the established standards, mouse operations, icon menus, etc. The demos include Probability Density Function (PDF), and Cumulative Distribution Function (CDF), normal, lognormal, exponential, Rayleigh and Rician random variables. The package also demonstrates how to generate the desired random sequence, how to examine its characteristics and how to estimate its PDF and CDF.