Summation of random errors having normal, uniform, and arcsine distribution laws

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

Introduction. The article develops a dynamic expert system (ES) based on a model of a system for monitoring the technical condition of signal generators in telecommunication systems. In the proposed system, decision-making is based on a constant analysis of the parameters of the time-frequency parameters of signals generated by generators in infotelecommunication systems. The purpose of the work is to develop an expert system based on a model of a system for monitoring the technical condition of signal generators in infotelecommunication systems. Research results: In the course of the work, an analysis of expert systems was carried out, a dynamic expert system was selected from the real-time communication classification, the structure of the expert system was proposed, each of its elements was considered, an expert system was developed that allows predicting changes in the parameters of transmission channel generators in telecommunication systems, and the resulting expert system was tested based on two assumptions, which allows us to draw conclusions about the correctness of the development. For this purpose, the result of processing the input data is given based on two assumptions – the law of normal distribution and the uniform distribution law. On the basis of this, it can be concluded that the proposed developed version of the ES construction and the software product implementing it is working. The expert system makes it possible to analyze incoming data in the current time, as well as generate an output.

The limit measurement errors in the geodesy are determining for the normal distribution law, But the errors, exceeding the limits are throw aside and accidental errors are obey the cut law. Cut laws are not study enough. The article is leading the results of calculate the probabilities hit accidental measurements errors for intervals σ, 2σ, 2,5σ, 3σ for cut normal and cut logical distribution laws. The analysis was accomplished.The probabilities of the cut normal and normal laws is different immaterial. 0,043 - maximum of the different. The normal law replace to the cut normal law is not recommending. Probabilities of the cut law may to use, when it is allowing to avoid the repetition of the field measurements.

The article discusses the influence of the stator winding temperature oscillation on the insulation-aging rate of the asynchronous motors stator winding in random mode loading. It is shown that  distribution laws of the losses value in motor and torque value are very essentially different, that must be considered when conducting research. With using a thermodynamic model of the totally enclosed fan cooled asynchronous motor, was evaluate the influence of winding temperature fluctuations on the accelerating reducing the thermal resource of insulation for uniform and normal distribution laws of the load. It is shown that under certain parameters of the random mode insulation durability can significantly decrease because of nonlinear coupling of thermal aging rate of the insulation with temperature, wherein the widely used average loss method gives the evaluation of the motor thermal state with a substantial error.

The performed study is the result of collecting statistical data on failures of port transshipment and floating cranes. It is found that many accidents of components and parts of crane mechanisms can be associated with an erroneous determination of design loads. In order to increase the strength reliability, numerous field tests were carried out in river and sea ports under various modes of transshipment for different types of cargo. Tests have shown that operational loads are random in nature, and their laws of distribution over time of individual crane mechanisms can be approximated by truncated normal Gauss laws, accounting for which at the stage of new design developments, as shown by the practice of performing strength fatigue calculations, is not possible. In the article, in order to use the obtained laws at the design stage, it is proposed to replace the representation of operational loads with initial moments that are common to the entire crane mechanism, and transfer individual characteristics consideration such as stress concentration coefficients, fatigue curve parameters, operating time or test base, to subsequent stages of machine components development. For practical testing of the operational loads representation by the normal Gauss laws, a critical analysis of their initial moments, which makes it possible to identify inconsistencies in the representations of variable loads proposed for strength calculation, is given. For the corrected truncated normal load distribution law, the initial moments, the value of the order of which corresponds to the formulas structure used in calculating fatigue resistance at variable load amplitudes (in particular, under alternating loading modes), are determined. The presentation of corrected normal distribution laws with initial moments complements the normative documents used in practice when performing strength calculations under variable loading conditions, significantly simplifies calculations and makes it possible to take it into account at the stage of new design developments in accordance with the formulas established in the current normative documents.

Warning and prevention of natural emergencies is based on the analysis and forecasting of threats of emergency situations in the state. At the same time, the forecasting of natural emergencies should be aimed at regulating natural safety in the country, assessing the threats of emergencies and their prevention. Prevention of natural emergencies of various natures, minimization of their consequences is a component of creating safe conditions for the individual and society as a whole, an integral part of the state policy of national security of the country. Ensuring safety in emergency situations requires reliable functioning of the emergency response system adequate to the level and nature of threats. The effectiveness of planning and implementing measures to prevent natural emergencies is determined by the quality of forecasting the threats of their occurrence Since the processes of occurrence of natural emergencies are of a random nature, knowledge of the law of distribution of the random variable “number of emergencies” in the state on the basis of statistical data acquires importance. The article presents the results of checking the compliance of statistical data on natural emergencies in the state with the normal distribution law. Based on the processing and analysis of monitoring data for natural emergencies, a hypothesis was put forward about the normal law of distribution of the random variable the «number of natural emergencies» in the state in accordance with the available information. Pearson’s goodness-of-fit test was chosen to test the proposed hypotheses. An assessment was made in accordance with the chosen criterion of the hypothesis about the agreement of the sample distribution with the law of normal distribution. Asa result of the research, it was found that the hypothesis put forward does not contradict the statistical data of monitoring natural emergencies over a certain period of time. The parameters of the established distribution law were evaluated.

Modern livestock development trends in the agricultural production structure include the use of feed mixers and distribution equipment. However, a stream of malfunctions and failures is reported for mixer-feeders due to factors that are specific to agricultural enterprises in Ukraine, which reduce the operational reliability of feed mixers and distributors. The study aims to evaluate the quantitative indicators of the in-service reliability of mixer-feeders and develop methods to improve them. The reliability test plan [NMT] was used, which implies a certain number of research objects (N), objects undergoing repair in case of performance loss (M), and tests terminated when the operating time (T) is reached. Initial information was collected and processed, and empirical data on the time between failures of KUHN PROFILE 12.2 DS and PROFILE 14.2 DS mixer-feeders in Ukraine were analysed. The analysis established that the faulty condition of mixers-feeders in the initial operation period is caused by the second complexity group failures. Failures related to the loss of the grinding and mixing mechanism, which limits the average time between failures of the feed mixture and distribution equipment, were identified among the second complexity malfunction group. A statistical analysis of the mixer-feeder failure occurrence was carried out. The failure probability of the grinding-mixing mechanism of the feed mixer was determined to be within the normal distribution law. The main parameters of the given theoretical distribution law were determined by calculation results. The calculations determined that the average time between failures is 3152.0 motor-hours; the standard deviation is 902.6 motor-hours; and the coefficient of variation is 0.40. An average time between failures of the study objects was proved to be consistent with the normal distribution law following Kolmogorov’s agreement criterion. Calculations determined that the average reliability index value, according to the normal distribution law, is within the confidence intervals: 1964.2 motor-hours – lower confidence limit; 4339.8 motor-hours – upper confidence limit. The obtained results may be used to improve the mixer-feeder operational reliability

Silicon carbide black grinding powders 54CF180-F36, mass-produced at Volzhsky Abrasive Plant are studied. Powders are screened into fractions according to GOST R 52381. Measurement of a grain width of each fraction was carried out by means of a photome-tric method using specially developed software. For the vast majority of fractions the grain width follows the normal distribution law. The deviation of the grain width from the normal law can be observed at the frac-tions settled on the second sieve and the pallet. With decreasing the nominal mesh size of a pass sieve in each grit size the arithmetic mean of grain width of the main fraction decreases, dispersion reduces from the fraction settling on the second sieve to the fourth and increases at the fraction settling on the pallet. In case of the identical mesh size of a pass sieve the mean and dispersion of grain width fraction increase with grinding powder grit increase. For each grit size the functional dependences of the mean and dispersion of grain width fraction on a mesh size of a pass sieve are established.

The purpose of the work is to increase the chi-square test power by analyzing the spectrum of output states of this test discrete form. The refusal from a hypothesis of a spectrum continuity of a chi-square molecule output states allows you to observe 22 spectral lines of different probability amplitudes on a sample of 11 experiments. It is proposed to obtain a mode of support for the quantum superposition of the output states of the chi-square molecule through the random selection of 11 experiments from the general population of 21 experiments. This makes it possible to obtain about 200,000 different states of the molecule, which is quite sufficient to estimate the 8-level states of the probability amplitude of 22 output spectral lines. In the case of the histogram with 6 intervals, the hypothesis of a continuous spectrum of the chi-square test for a sample of 11 experiments does not allow us to distinguish between normal and uniform laws of data distribution with a confidence probability acceptable for practice. The situation changes dramatically if the spectrum discrete nature of the chi-square molecule states is taken into account. It is possible to obtain significantly different digital portraits of the normal and uniform data distribution. Their differences are about 40 bits as per the Hamming distance. The chi-square molecule can be interpreted as a certain square neuron with a Gammerstein structure. Simulation of Schrödinger equations on an ordinary computer has exponential computational complexity in relation to the number of electrons (degrees of freedom or experiments). For this reason, the reproduction of quantum computations on 22 qubits and more within the framework of quantum mechanics on an ordinary computer is technically difficult to perform. A completely different situation arises with the support of quantum superposition of the chi-square molecule. The support of quantum superposition of the chi-square molecule has linear computational complexity. As a consequence, it is possible to implement 22-qubit, 222-qubit, 2222-qubits, and higher quantum computations on an ordinary computer.

In the recent work devoted to the calculation of the rovibrational line list of the CO molecule [G. Li et al., Astrophys. J., Suppl. Ser. 216, 15 (2015)], rigorous validation of the calculated parameters including intensities was carried out. In particular, the Normal Intensity Distribution Law (NIDL) [E. S. Medvedev, J. Chem. Phys. 137, 174307 (2012)] was employed for the validation purposes, and it was found that, in the original CO line list calculated for large changes of the vibrational quantum number up to Δn = 41, intensities with Δn > 11 were unphysical. Therefore, very high overtone transitions were removed from the published list in Li et al. Here, we show how this type of validation is carried out and prove that the quadruple precision is indispensably required to predict the reliable intensities using the conventional 32-bit computers. Based on these calculations, the NIDL is shown to hold up for the 0 → n transitions till the dissociation limit around n = 83, covering 45 orders of magnitude in the intensity. The low-intensity 0 → n transition predicted in the work of Medvedev [Determination of a new molecular constant for diatomic systems. Normal intensity distribution law for overtone spectra of diatomic and polyatomic molecules and anomalies in overtone absorption spectra of diatomic molecules, Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka, 1984] at n = 5 is confirmed, and two additional "abnormal" intensities are found at n = 14 and 23. Criteria for the appearance of such "anomalies" are formulated. The results could be useful to revise the high-overtone molecular transition probabilities provided in spectroscopic databases.

The article deals with matters concerned with the problem of determining the statistical characteristics of variable parameters (the variation range and distribution law) in analyzing the uncertainty and sensitivity of calculation results to uncertainty in input data. A comparative analysis of modern approaches to uncertainty in input data is presented. The need to develop an alternative method for estimating the uncertainty of model parameters used in thermal–hydraulic computer codes, in particular, in the closing correlations of the loop thermal hydraulics block, is shown. Such a method shall feature the minimal degree of subjectivism and must be based on objective quantitative assessment criteria. The method includes three sequential stages: selecting experimental data satisfying the specified criteria, identifying the key closing correlation using a sensitivity analysis, and carrying out case calculations followed by statistical processing of the results. By using the method, one can estimate the uncertainty range of a variable parameter and establish its distribution law in the above-mentioned range provided that the experimental information is sufficiently representative. Practical application of the method is demonstrated taking as an example the problem of estimating the uncertainty of a parameter appearing in the model describing transition to post-burnout heat transfer that is used in the thermal–hydraulic computer code KORSAR. The performed study revealed the need to narrow the previously established uncertainty range of this parameter and to replace the uniform distribution law in the above-mentioned range by the Gaussian distribution law. The proposed method can be applied to different thermal–hydraulic computer codes. In some cases, application of the method can make it possible to achieve a smaller degree of conservatism in the expert estimates of uncertainties pertinent to the model parameters used in computer codes.

Some Thoughts About Data Type, Distribution, and Statistical Significance

Актуальность темы исследования обусловлена тенденцией к применению постоянных магнитов в конструкциях электромеханических преобразователей, которая выражается в многополюсном исполнении синхронных магнитоэлектрических машин, работающих в генераторном и двигательном режимах [1, 2]. В статье представлено совершенствование методики идентификации постоянных магнитов, приведенной в [3] на основе исследования закона распределения магнитных сил взаимодействия между опытными образцами постоянных магнитов и магнитным отражателем на соответствие нормальному закону распределения по критерию согласия Пирсона. Показана возможность использования методов и свойств нормального закона распределения при решении вопроса идентификации постоянных магнитов с одинаковой объемной намагниченностью и выборки постоянных магнитов из опытной партии для разработки и создания магнитоэлектрических машин различного назначения.

In time series analysis, especially in non-linear time series, a school appeared after 1963. The classical school argued share price variations obey the normal distribution law and are random, in which case observations are identically distributed and independent. Although Brown (1828), Bachelier (1900), and Einstein (1905) created the background, the well-known AR(IMA) models were invented by Yule (1927). The modern school argued that time series appear to be random with a behavior under determinism, are subject to power law, and have a long run memory. Recent (nonlinear) work has not related to shipping freight derivatives, but to ship prices and freight markets. Derivatives are known as a shipping risk management means. According to the classical theory, derivative value should relate to volatility and reward, and volatility relates to standard deviation, provided the randomness of freight rates, for freight market risk management. That, to the shipping industry and its bankers, risk management is an alien concept, was written about by Stokes (1997). Stokes likened the typical shipowner to a speculator or a gambler who enjoys making large bets, and, in the hope of winning an enormous prize, risks liberal amounts of money. The authors' personal experience is that Greek shipowners, at least, are conservative. There are times when it is true, however, that in a three-year (ship) operation, more money comes from a single asset's sale than its use. Both tremendous failures and unbelievable fortunes have been created by the shipping industry, and the bankers pick up the pieces at day's end. People are attracted to the industry by massive successes, such as those seen in recent years, and repelled by failures. That risk must be managed is of no doubt, for it certainly exists. Understanding relevant market natures allow this best to be achieved. The authors demonstrate that shipping freight derivatives are persistent, while freight markets are anti-persistent, rather than random. Interaction between two opposing forces seems to occur through this phenomenon. Shipping freight market analysis must be done in a general finance context. Mandelbrot-Hudson (2004) approach applications suggests that a catastrophe's real chances of happening range between 1/10 and 1/30, or, in other words, it is much greater than what would be indicated by random walk.

The paper analyzes the homogeneity of the stochastic structure of the samples of oil flax fibres of the Ukrainian and Polish selections. The distribution laws of fibres as mixtures of the normal and logarithmically normal distribution laws are identified. The general probability distribution law of fibre lengths of oil flax of the Ukrainian selection is a mixture of four logarithmically normal laws of distribution, and the general probability distribution law of fibre lengths of oil flax of the Polish selection is a mixture of a normal distribution law and three logarithmically normal laws of distribution. The parameters of the relevant components of the common distribution laws are found as a result of solving the problem of nonlinear optimization using the method of generalized lowering gradient that is implemented in Excel tabular processor. The adequacy of the built nonlinear models of the distribution laws is proved by estimating the value of the determination index. The comparative analysis of distribution laws of general totalities, from which the oil flax fibre samples of the Ukrainian and Polish selections are obtained, is conducted by the Kolmogorov-Smirnov statistical criterion. This criterion allows us to test and reject the hypothesis about belonging of two independent samples to the same general totality. The analysis of the general distribution laws shows that the oil flax fibres of the Ukrainian selection have a more pronounced tendency to stratification by a fibre length. Such a fibre is easily divided into factions. The oil flax fibres of the Polish selection have a larger part of short and extra-long fibres. Thus, it is clear that raw materials of the Ukrainian selection are more suitable for the production of non-woven materials of different target purposes, and the Polish selection - for the production of ropes, cords, etc. The obtained results can be used by specialists at processing enterprises for a reasonable choice of raw materials, optimization of processing modes and improving ready-made product quality.