Setting the stage: uniform law

2021 ◽  
pp. 11-59
Keyword(s):  
2017 ◽  
Vol 21 (3) ◽  
pp. 458-461
Author(s):  
Eric Clive
Keyword(s):  

Author(s):  
T. V. Oblakova

The paper is studying the justification of the Pearson criterion for checking the hypothesis on the uniform distribution of the general totality. If the distribution parameters are unknown, then estimates of the theoretical frequencies are used [1, 2, 3]. In this case the quantile of the chi-square distribution with the number of degrees of freedom, reduced by the number of parameters evaluated, is used to determine the upper threshold of the main hypothesis acceptance [7]. However, in the case of a uniform law, the application of Pearson's criterion does not extend to complex hypotheses, since the likelihood function does not allow differentiation with respect to parameters, which is used in the proof of the theorem mentioned [7, 10, 11].A statistical experiment is proposed in order to study the distribution of Pearson statistics for samples from a uniform law. The essence of the experiment is that at first a statistically significant number of one-type samples from a given uniform distribution is modeled, then for each sample Pearson statistics are calculated, and then the law of distribution of the totality of these statistics is studied. Modeling and processing of samples were performed in the Mathcad 15 package using the built-in random number generator and array processing facilities.In all the experiments carried out, the hypothesis that the Pearson statistics conform to the chi-square law was unambiguously accepted (confidence level 0.95). It is also statistically proved that the number of degrees of freedom in the case of a complex hypothesis need not be corrected. That is, the maximum likelihood estimates of the uniform law parameters implicitly used in calculating Pearson statistics do not affect the number of degrees of freedom, which is thus determined by the number of grouping intervals only.


The author, considering that the careful dissections of Meckel and Cuvier have fully established the universal existence of a thyroid gland in the whole of the class Mammalia, proceeds to consider the comparative anatomy of this organ in the remaining classes of vertebrated animals. His dissections of birds have included all the orders, and, in most instances, several families from each: he has never failed to find in them a thyroid gland, and, with the aid of the microscope, to recognise its peculiar structure; he presumes, therefore, that it is universally present in that class of animals. He has also detected the presence of this organ in reptiles of every order; although generally either wholly overlooked by anatomists, or mistaken for the thymus. Descriptions are here given of its appearance, position and structure in different families of Chelonia, Sauria, Ophidia and Batrachia. In the class of Fishes, it is by no means universally or even generally present. The author has found it in the carp, anableps, pike, exocetus, cod, haddock, whiting, eel, sturgeon, callorhynchus, shark and skate, and perhaps in the lamprey. On the other hand, it appears to be absent in the perch, mullet, gurnard, mackerel, tench, salmon, trout, herring, plaice, halibut, turbot, sole, cyclopterus, gymnotus and balistes. The general conclusion which the author deduces from his researches is, that the distribution of the thyroid gland is regulated by a simple and uniform law; being dependent on the existence or nonexistence of another organ with which its presence alternates, and which, in many fishes, assumes the form of a minute supplementary gill, the vessels of which communicate, on the one hand, with the systemic veins about the base of the cranium, and on the other, by a single long trunk with the first branchial vein.


1907 ◽  
Vol 1 (3) ◽  
pp. 565-578 ◽  
Author(s):  
Simeon E. Baldwin

The pre-Tridentine œcumenical councils of the Roman Catholic church were, as Dr. Francis Wharton has well remarked, international congresses, working toward the establishment of a uniform law for the civilized world. It was a law confined to one set of subjects; but among them were those having to do with the family relation, and which were therefore of the first importance to human society. Each nation of Christendom was represented in these gatherings by its sovereign or political delegates, as well as by its bishops, and it was for each nation, acting through its political departments, to ratify or reject such rules or laws in these respects as the council might propose.The representation of political sovereignty in the Council of Trent was slight, and in the only œcumenical council since called by Eome — that of the Vatican — it was wholly wanting; Bavaria being the only power (though all European cabinets were consulted) which intimated a willingness to send an official delegate.


2006 ◽  
Vol 43 (4) ◽  
pp. 1107-1118 ◽  
Author(s):  
Alexander D. Kolesnik

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.


2021 ◽  
Vol 4 (1) ◽  
pp. 143-165

The article aims to examine one of the elements of the formal mechanism of maintaining court practice unity in criminal proceedings of Ukraine and European countries – referring a case to the highest division of the Supreme Court. Similar to the Ukrainian criminal procedure legislation, the grounds for referring a criminal case and the procedure of its application are provided in the legislation of Estonia, Italy and Lithuania. At the same time, the Ukrainian legislator has established a number of special features, however, the wording of the relevant articles of the Criminal Procedure Code of Ukraine is not perfect. The article provides answers to such questions as how forceful the provisions of criminal procedure legislation of Ukraine are, to what extent of effectiveness the Supreme Court exercises its legal authority regarding the unity of court practice in criminal proceedings, and whether the controversies in legal positions of the structural divisions of the Supreme Court have been successfully avoided. In order to achieve the stated aims, parts 2 and 3 are devoted to the examination of the grounds for referring a case in criminal proceedings of Ukraine and European countries. Part 4 outlines the shortcomings of the content of some articles of the Criminal Procedure Code of Ukraine concerning the procedure of the referral of a criminal case to the highest division of the Supreme Court. Part 5 provides the analysis of the validity of decisions made by the boards of judges at the Supreme Court on the referral of criminal proceedings to its higher judicial divisions – the joint chamber of the Criminal Cassation Court and the Grand Chamber of the Supreme Court. On the basis of the study of the judgements of boards, the judicial chambers of the Criminal Cassation Court and the Grand Chamber of the Supreme Court, in part 6 the question is answered on whether the Supreme Court of Ukraine managed to perform its duty on the assurance of court practice unity in such an area as criminal proceedings. Keywords: exclusive legal problem, development of law, formation of uniform law enforcement practice, the Supreme Court, criminal proceedings, Ukraine.


Sign in / Sign up

Export Citation Format

Share Document