Statistical Distribution Theory

Exercises and solutions for a second year statistics course.

On the Statistical Distribution of Object-Oriented System Properties

The statistical distributions of different software properties have been thoroughly studied in the past, including software size, complexity and the number of defects. In the case of object-oriented systems, these distributions have been found to obey a power law, a common statistical distribution also found in many other fields. However, we have found that for some statistical properties, the behavior does not entirely follow a power law, but a mixture between a lognormal and a power law distribution. Our study is based on the Qualitas Corpus, a large compendium of diverse Java-based software projects. We have measured the Chidamber and Kemerer metrics suite for every file of every Java project in the corpus. Our results show that the range of high values for the different metrics follows a power law distribution, whereas the rest of the range follows a lognormal distribution. This is a pattern typical of so-called double Pareto distributions, also found in empirical studies for other software properties.

The statistical distribution of ocean wave forces on vertical piling /

"July 1965."

Tables of the statistical distribution of ocean wave forces and methods of estimating drag and mass coefficients /

"October 1967."

Statistical distribution, host for encrypted information

The statistical distribution, when determined from an incomplete set of constraints, is shown to be suitable as host for encrypted information. We design an encoding/decoding scheme to embed such a distribution with hidden information. The encryption security is based on the extreme instability of the encoding procedure. The essential feature of the proposed system lies in the fact that the key for retrieving the code is generated by random perturbations of very small value. The security of the proposed encryption relies on the security to interchange the secret key. Hence, it appears as a good complement to the quantum key distribution protocol. © 2005 Elsevier B.V. All rights reserved.

POISSON DISTRIBUTION TO ANALYZE NEAR-THRESHOLD MOTOR EVOKED POTENTIALS

Motor unit action potentials (MUAPs) evoked by repetitive, low-intensity transcranial magnetic stimulation can be modeled as a Poisson process. A mathematical consequence of such a model is that the ratio of the variance to the mean of the amplitudes of motor evoked potentials (MEPs) should provide an estimate of the mean size of the individual MUAPs that summate to generate each MEP. We found that this is, in fact, the case. Our finding thus supports the use of the Poisson distribution to model MEP generation and indicates that this model enables characterization of the motor unit population that contributes to near-threshold MEPs. Muscle Nerve 42: 825-828, 2010

Statistical assessment of dataset shift and model portability in multi-angle in-track image acquisitions

In this study we propose an evaluation of the angular effects altering the spectral response of the land-cover over multi-angle remote sensing image acquisitions. The shift in the statistical distribution of the pixels observed in an in-track sequence of WorldView-2 images is analyzed by means of a kernel-based measure of distance between probability distributions. Afterwards, the portability of supervised classifiers across the sequence is investigated by looking at the evolution of the classification accuracy with respect to the changing observation angle. In this context, the efficiency of various physically and statistically based preprocessing methods in obtaining angle-invariant data spaces is compared and possible synergies are discussed.

Power law distribution in education: Effect of economical, teaching, and study conditions in university entrance examination

We studied the statistical distribution of student's performance, which is measured through their marks, in university entrance examination (Vestibular) of UNESP (Universidade Estadual Paulista) with respect to (i) period of study - day versus night period (ii) teaching conditions - private versus public school (iii) economical conditions - high versus low family income. We observed long ubiquitous power law tails in physical and biological sciences in all cases. The mean value increases with better study conditions followed by better teaching and economical conditions. In humanities, the distribution is close to normal distribution with very small tail. This indicates that these power law tails in science subjects axe due to the nature of the subjects themselves. Further and better study, teaching and economical conditions axe more important for physical and biological sciences in comparison to humanities at this level of study. We explain these statistical distributions through Gradually Truncated Power law distributions. We discuss the possible reason for this peculiar behavior.

Gradually truncated log-normal in USA publicly traded firm size distribution

We study the statistical distribution of firm size for USA and Brazilian publicly traded firms through the Zipf plot technique. Sale size is used to measure firm size. The Brazilian firm size distribution is given by a log-normal distribution without any adjustable parameter. However, we also need to consider different parameters of log-normal distribution for the largest firms in the distribution, which are mostly foreign firms. The log-normal distribution has to be gradually truncated after a certain critical value for USA firms. Therefore, the original hypothesis of proportional effect proposed by Gibrat is valid with some modification for very large firms. We also consider the possible mechanisms behind this distribution. (c) 2006 Published by Elsevier B.V.

Power-law distribution in a learning process: competition, learning and natural selection

In the present work, we propose a model for the statistical distribution of people versus number of steps acquired by them in a learning process, based on competition, learning and natural selection. We consider that learning ability is normally distributed. We found that the number of people versus step acquired by them in a learning process is given through a power law. As competition, learning and selection is also at the core of all economical and social systems, we consider that power-law scaling is a quantitative description of this process in social systems. This gives an alternative thinking in holistic properties of complex systems. (C) 2004 Elsevier B.V. All rights reserved.

Power law distribution in education: University entrance examination

We studied the statistical distribution of candidate's performance which is measured through their marks in university entrance examination (Vestibular) of UNESP (Universidade Estadual Paulista) for years 1998, 1999, and 2000. All students are divided in three groups: Physical, Biological and Humanities. We paid special attention to the examination of Portuguese language which is common for all and examinations for the particular area. We observed long ubiquitous power law tails in Physical and Biological sciences. This indicate the presence of strong positive feedback in sciences. We are able to explain completely these statistical distributions through Gradually Truncated Power law distributions which we developed recently to explain statistical behavior of financial market. The statistical distribution in case of Portuguese language and humanities is close to normal distribution. We discuss the possible reason for this peculiar behavior.

The Kumaraswamy Gumbel distribution

The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. We propose a generalization-referred to as the Kumaraswamy Gumbel distribution-and provide a comprehensive treatment of its structural properties. We obtain the analytical shapes of the density and hazard rate functions. We calculate explicit expressions for the moments and generating function. The variation of the skewness and kurtosis measures is examined and the asymptotic distribution of the extreme values is investigated. Explicit expressions are also derived for the moments of order statistics. The methods of maximum likelihood and parametric bootstrap and a Bayesian procedure are proposed for estimating the model parameters. We obtain the expected information matrix. An application of the new model to a real dataset illustrates the potentiality of the proposed model. Two bivariate generalizations of the model are proposed.

Statistical distributions obtained from the compression of monodisperse, soft and frictionless particles

The cyclic compression of several granular systems has been simulated with a molecular dynamics code. All the samples consisted of bidimensional, soft, frictionless and equal-sized particles that were initially arranged according to a squared lattice and were compressed by randomly generated irregular walls. The compression protocols can be described by some control variables (volume or external force acting on the walls) and by some dimensionless factors, that relate stiffness, density, diameter, damping ratio and water surface tension to the external forces, displacements and periods. Each protocol, that is associated to a dynamic process, results in an arrangement with its own macroscopic features: volume (or packing ratio), coordination number, and stress; and the differences between packings can be highly significant. The statistical distribution of the force-moment state of the particles (i.e. the equivalent average stress multiplied by the volume) is analyzed. In spite of the lack of a theoretical framework based on statistical mechanics specific for these protocols, it is shown how the obtained distributions of mean and relative deviatoric force-moment are. Then it is discussed on the nature of these distributions and on their relation to specific protocols.

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Matrix-Variate Statistical Distributions and Fractional Calculus

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthday