Distributions Of Random Variables Research Articles

The -User DM Broadcast Channel With Two Groupcast Messages: Achievable Rate Regions and the Combination Network as a Case Study

A novel class of achievable rate regions is obtained for the general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -receiver discrete memoryless broadcast channel over which two groupcast messages are to be transmitted, with each message required by an arbitrary group of receivers. The associated achievability schemes are parameterized by an expansion of the message set which then determines how random coding techniques are employed. These techniques include generalized versions of up-set message-splitting, the generation of possibly multiple auxiliary codebooks for certain compositions of split messages using superposition coding with subset inclusion order, partial interference decoding at all receivers in general, joint unique decoding at receivers that desire both messages, and non-unique or indirect decoding at receivers that desire only one of the two messages. The generality of the proposed class of schemes implies new achievable rate regions for problems previously not considered as well as those that were studied before, with specific members of that class having rate regions that coincide with previously found capacity regions for special classes of broadcast channels with two private or two nested groupcast messages, wherein the group of receivers desiring one message is contained in that desiring the other. Moreover, new capacity results are established for certain partially ordered classes of broadcast channels for a class of two non-nested groupcast messages. To further show the strength of the proposed achievable rate regions we consider the so-called combination network as a test case. When specialized to the combination network, some members of the class of inner bounds are shown, via converse results, to result in the capacity region when the two messages are (a) intended for two distinct sets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K{-}1$ </tex-math></inline-formula> receivers each and (b) nested, in which one message is intended for one or two (common) receivers and both messages are intended for all other (private) receivers. In the latter two nested messages cases, we hence recover, in a top-down manner, previous results by Bidokhti, Prabhakaran, and Diggavi, obtained therein using lower complexity network coding schemes based on rate-splitting and linear superposition coding but tailored to the combination network, while in the first case we obtain a new capacity result for a non-nested message set, which was hitherto unknown. Furthermore, we show the achievability of rate pairs in two interesting examples of combination networks, with three and four common receivers each. These examples were proposed in the previous literature to show the sub-optimality of the aforementioned rate-splitting and linear superposition coding scheme, and hence to motivate the additional consideration of a pre-encoding technique and a block-Markov linear superposition coding for the combination network, with the latter then lifted to the general broadcast channel. Our results suggest that the proposed framework here for the general broadcast channel when specialized to the combination network is strong enough to incorporate the enhancements afforded by those two latter techniques, thereby implying that perhaps block-Markov superposition coding is not necessary in the general broadcast channel. Moreover, there is a trade-off between the complexity of the coding scheme within the class of schemes we propose when applied to the combination network and that of the determination of the distribution of the auxiliary random variables and the encoding function that achieve the capacity region. This may have interesting implications for the general broadcast channel as well.

Signaling-based Robust Incentive Designs with Randomized Monitoring

Incentive design problems entail hierarchical decision-making where a leader crafts a strategy to induce a desired response from a follower. Such dynamic games with decentralized information structures have been well-studied under three assumptions-the leader must have access to the follower's observations, actions, and the objective function. Lack of knowledge on any of these can potentially lead to performance loss for the leader. In this paper, we first study a setup where the leader observes the follower's action through a random monitoring channel and learns about the follower's observation through a follower-designed signal. In this setup, we establish the existence of a signaling-based incentive equilibrium strategy for the leader that induces honest reporting and desired control response from the follower. Then, we study a setting, where the follower's costs are parametric, but the parameters are not known to the leader. We construct an incentive strategy that reduces the sensitivity of the leader's performance to uncertainty in the parameter, close to an initial estimate. More generally, for the case when the leader's knowledge about the follower's cost and distributions of cost-relevant random variables is inaccurate, we establish the existence of a robust incentive equilibrium strategy that bounds the performance loss from the inaccuracy in the model.

Hidden Markov Model and States Prediction of an Autonomous Wind-Diesel Complex

The problem of assessing the reliability and analyzing the functioning of an autonomous winddiesel complex, consisting of a wind power plant, working and standby diesel generators, an inverter and a storage battery, is considered. First, a semi-Markov model of an autonomous wind-diesel complex is built, which makes it possible to calculate the stationary and temporal reliability characteristics. Then, on its basis, a hidden Markov model is developed, which is used to solve the problems of predicting and evaluating its characteristics, taking into account the given parameters and the signal vector. The results of the study are obtained in a general form and are invariant with respect to the laws of distribution of random variables describing the elements of an autonomous wind-diesel complex. They allow you to simulate the functioning of the system under various distribution laws, based on statistical data, without modifying the model itself.

Вероятность появления повреждений и отказов на трубопроводах оросительных систем

Purpose: to process data on damage and failures on the irrigation pipes to assess the technical condition of the pipeline, to define its performance reliability, terms of its failure-free operation, to determine the time after which the pipeline needs repairing, reconstruction or replacing of certain elements. Materials and methods: the analysis of statistical data on defects and damage on the Engels irrigation system pipeline was carried out; mathematical processing of data on pipe damage and failures was carried out using the methods of mathematical statistics with their consideration as random variables: first, the number of dam-age and failures was calculated, then the empirical distribution of random variables and the empirical probabilities of their distribution were compiled; the expectation value of damage and failures occurrence on the pipeline is calculated. Results: according to the field observa-tions, a list of defects and damage on this pipeline recorded in the corresponding observation log was compiled, and possible causes of their occurrence presented in tabular form and figures were given; based on the results of calculations, a histogram and theoretical distribution curves for the expectation values of damage and failures occurrence on the pipeline were constructed. Conclusion: the probability of the first damage and failures occurrence after 14 years of pipe operation has been determined, further damage and failures increase, and by the 17th–18th year in order to prevent damage and failures, it is necessary to repair the pipeline, joints and stop valves due to corrosion, cracks, chips, dents and other damage.

STUDY OF REGRESSION MODELS WITHOUT A FREE TERM

Purpose. The aim of the article is to use the analogue of the coefficient of determination of multiple regression for a classical regression model without a free term and constructing the appropriate Fisher and Student statistics for econometric analysis. Methodology of research. Methods of general scientific and empirical techniques based on a systemic approach were used in the process of research. In addition, the following methods of scientific knowledge were used to solve the tasks: methods of analysis and observation - to identify the main social factors influencing the level of crime; methods of matrix theory, theory of probabilities and mathematical statistics - for checking the laws of distribution of random variables, building criteria for the significance of the regression equation and its coefficients. Findings. It was studied how the absence of a free term in multiple linear regression affects the Student and Fisher statistics when studying the significance of regression coefficients and the regression equation as a whole, respectively. The obtained results indicate that the introduced definitions affect the adjusted coefficient of multiple regression and the F-statistic, but do not change the standard error of the regression, including the Student's statistics. The need to calculate qualitative indicators of regression models in accordance with derived formulas and algorithms, and to be critical of the final numerical results obtained when using computer statistical packages, is substantiated. Originality. A comprehensive study of linear regression models was conducted, which takes into account the effect of the absence of a free term when evaluating the quality of empirical models. Practical value. The obtained research results allow avoiding calculation errors and acceptance of implausible conclusions, which occur in the practice of qualitative analysis of linear regression models without a free term. Key words: multiple regression model, free term, method of least squares, Student's test, Fisher's test.

The Application of Stochastic Seismic Inversion Method of Geostatistical Inversion in Putaohua X Area

Stochastic seismic inversion is an important part of geostatistical inversion method, and is the combination of seismic inversion and sequence simulation. In the process of random seismic inversion, firstly, the histogram distribution of random variables is counted, the variation function of random variables is calculated, and the variation range of random variables in different directions is determined. Then, for each implementation, the seismic traces are synthesized at the seismic channel position according to the calculated reflection coefficient, and the error between the synthetic traces and the original seismic traces is compared, and the inversion results are output after the required accuracy is reached. Geostatistical inversion can improve the vertical and horizontal resolution of the reservoir and accurately describe the spatial distribution of the reservoir. Taking the Heidimiao oil layer in the west slope area of Daqing Placanticline as an example, this paper expounds the application of geostatistical inversion in the work area as a whole. Combined with the results of seismic attribute analysis and well logging curve preprocessing, the inversion results formed by accurate time-depth calibration and other processes of reservoir sensitivity parameter analysis provide a basis for optimizing the target well location in the work area.

STUDY OF DURABILITY IN THE CONNECTION “CONE-TUNGSTEN CARBIDE INSERT” FOR TENSION DEPENDENCE

During and after the technological operation of pressing these factors significantly determine the residual stress states of tungsten carbide inserts, as well as in the areas near the holes of the cones’ crowns of the cone. The stressed state of the inserts and the magnitude of the stress concentration determine their fatigue strength of them and the durability of the inserted carbide rock-blasting equipment of the cones in the area where the inserts the cone. Dimensional analysis was done for the inserts and holes in the cones of bit and it was established that size errors of component links of dimensional chains are distributed according to laws that are similar to the normal law of distribution of random variables. The test was carried out according to the criteria of Pearson and Kolmogorov. The influence of various random and systematic factors results to the dispersion of dimensions on the dimensional processing of conjugated surfaces. The amount of tension is a function of the dimensions for mating surfaces during assembly operations. Dispersion of dimensions for the component links of the dimensional chain of a press connection causes the formation of a practical field of dispersion of the closing link. Accuracy control for the closing link in the dimensional chain is impossible if there are no direct measurements of the constituent links. A mathematical model of the process of assembling "cone-tungsten carbide insert" joints was created. For this, the methods of mathematical statistics were applied in the study. It was established that the tension values have a greater influence on the force of pressing inserts into the holes of drill steels 14 NiCrMo1, when its hardness value is HRC 59-60 compared to HRC 48-50. This dependence has a linear character within the limits of the studied tension values. The established relationships make it possible to reasonably and most accurately form selective groups of inserts and mark holes for them. This method significantly reduces losses in the production of drilling tools.

TWO-SYMBOL SYSTEM OF ENCODING OF NUMBERS AND DISCRETE DISTRIBUTIONS OF RANDOM VARIABLES

We consider discrete distributions of random variables, defined by various two-symbol systems of encoding of real numbers (with zero and non-zero redundancy, with one and two bases, in particular with different sings), and study structural, topological, metric, and structurally fractal properties their point spectra. The general criterion for random variable with independent digits of two-symbol representation to have discrete distribution (analog of the P. L’ evi theorem for sum of random series with discretely distributed terms) is proved and properties of its spectrum are described. In the paper we study discrete distributions of values of functions of the Cantor type of a random continuously distributed argument.

Modification of a nonparametric method for testing the hypothesis of distributions of random variables

Modification of a nonparametric method for testing the hypothesis of distributions of random variables

Guarantees and risks in the technical regulation of transport construction

As a result of the analysis of the methods of quantitative and qualitative assessment of the degree of risk, the theoretical and probabilistic approach of Professor V.V. Stolyarov was chosen to assess technical and environmental risk, based on the probabilistic nature of the parameters studied, in relation to road facilities.The allocation of risks within functional subsystems makes it possible to more accurately determine the ways of risk management in practice. As a characteristic of the degree of danger of a road section, the total risk is used as a complex indicator that can occur on this section under the combined influence of all road parameters simultaneously. Along with the discussion of the concepts of risk, the concept of guarantee is introduced. The concepts of guarantees and risks are introduced on the basis of a random variable distribution graph using the example of a normal distribution. In addition to the technical component of the risk, its economic component is also considered.

Влияние облучения на заболеваемость атеросклерозом среди участников ликвидации последствий аварии на Чернобыльской атомной электростанции

The statistical relationship between radiation dose to the cohort of workers (liquidators), participated in mitigation of effects of the Chernobyl accident are considered in the paper. The follow-up data collected from 1992 to 2021 and accumulated in the National Radiation and Epidemiological Register (NRER) were used for analysis. The cohort of liquidators consisted of 67,304 males. The average radiation dose in the cohort was 0.13 Gy, and the average age at radiation exposure was 34 years. During the follow-up period 10,790 atherosclerosis cases (ICD-10, code 170) were detected for the first time. To identify statistical relationship between radiation dose and atherosclerosis incidence, general “data mining” approaches were used, that do not use hypotheses about the probabilistic laws of random variable distributions, that determine the relationship between radiation dose and atherosclerosis incidence. A group of liquidators with accumulated during the period of work individual radiation doses below 0.15 Gy was selected as the reference group. The maximum radiation dose in the studied cohort was 1.4 Gy. Statistically significant relationship between radiation doses and atherosclerosis incidence (ICD-10, code 170) were found: relative radiation risk RR=1.05 at 95% lower confidence bound 1.027. The relative risks (RR) for individual diagnoses are: RR=1.22 (95% CI lower bound – 1.20) for ICD-10 – I70.0 (aortic atherosclerosis); RR=1.03 (95% CI lower bound – 1.01); for ICD-10 – I70.2 (limb artery atherosclerosis); RR=1.06 (95% CI lower bound – 1.04) for ICD-10 – I70.9 (generalized and unspecified atherosclerosis). The RR increases with the age at diagnosis. The results obtained can be the basis for the design of radiation-epidemiological studies for the detailed quantitative study of the detected dose-effect relationships.

New proofs of Khinchin's law of large numbers and Lindeberg's central limit theorem –PDE's approach

&lt;p style='text-indent:20px;'&gt;Both Khinchin's law of large numbers (Khinchin's LLN) and Lindeberg's central limit theorem (Lindeberg's CLT) are fundamental results in probability theory. In this paper, we give the new proofs of these two theorems. A law of large numbers and a central limit theorem are proved for independent and non-identical distributed random variables. Indeed, these results include the Khinchin's LLN and Lindeberg's CLT. Our main tool is the viscosity solution theory of partial differential equation (PDE).&lt;/p&gt;

Distributional inequalities for noncommutative martingales

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.

Estimating the Parameters of Electric Energy Storages for Traction Substations

The article substantiates the relevance of using an electric energy storage in railway transport traction power supply systems for smoothing the traction substation load schedule taking the West Siberian railway as an example. The traction substation load schedule is analyzed with the consumed power values 30-min measurement intervals. The traction substation power consumption is estimated by a probabilistic method. The obtained analysis results were used to select the installed capacity of an electric energy storage for performing hourly load regulation and determine the traction substation average consumed power relative to which the load schedule will be smoothed. The hypothesis of the normal distribution of the traction substation load value is tested using application software packages for statistical data processing. The random variable typical distribution law most confidently reflecting the experimental data distribution pattern is determined. The energy storage’s selected power capacity and the traction substation average power consumption are used to perform a probabilistic assessment of the storage device exchange energy capacity. A procedure for selecting the storage device tentative exchange energy capacity is proposed.

On Polyhedral Coherent Risk Measures and Portfolio Optimization Problems

Introduction. The problem of decision-making under risk and uncertainty lies in the use of adequate criteria for assessing their optimality, in particular, in an adequate risk assessment. Various functions are known that are used as risk measures. For technical systems, the probability of an accident (failure) is used, in insurance – the probability of bankruptcy, in finance – Value-at-Risk, etc. At present, the concept of a coherent risk measure (CRM), in which its basic properties are postulated, is widely recognized. The paper considers CRMs and their subset, the polyhedral CRMs (PCMRs), which have attractive properties and contain a number of important risk measures. Such risk measures are well defined on complete information about the stochastic distributions of random variables. However, applications usually contain only partial such information from observational data. This only allows one to describe the stochastic distribution by an ambiguity set (AS). For such a case, robust PCMR constructions intended for risk assessment at AS are considered in the paper. The computation of such PCRM constructions in the form of linear programming problems (LP) is described. To demonstrate the use of the PCRM apparatus, the problems of portfolio optimization on reward-risk ratio are considered, where reward and risk are estimated by the average return and some PCRM respectively for known stochastic distributions, and by their robust constructions under uncertainty with AS. It is described how in both these cases the portfolio optimization problems are reduced to appropriate LP problems. The purpose of the paper is to describe the PCRM apparatus for assessing risks under uncertainty with AS and demonstrating the effectiveness of its application to linear problems on the example of portfolio optimization problems. Results. The use of the PCRM apparatus for the case of uncertainty with AS in the form of appropriate robust constructions and their application to portfolio optimization problems on reward-risk ratio is described. The conditions under which these portfolio problems are reduced to the corresponding LP tasks are formulated. Conclusions. The PCRM apparatus can be effectively applied to linear optimization problems under uncertainty with AS, which is demonstrated by the example of portfolio optimization problems. The reduction of portfolio problems to LP problems allows one to effectively solve them using standard methods. Keywords: coherent risk measure, polyhedral coherent risk measure, CVaR, ambiguity set, portfolio optimization, linear programming problem.

Cycle structure of random parking functions

We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper bound on the total variation distance between the joint distribution of cycle counts and independent Poisson random variables using a multivariate version of Stein's method via exchangeable pairs. Under a mild condition, the process of cycle counts converges in distribution to a process of independent Poisson random variables.

Assessment of the reliability of the operation of a sewage treatment plant using Monte Carlo simulation

The aim of the study was to model the operation of a wastewater treatment plant using the Monte Carlo method and selected probability distributions of random variables. Pollutant indices in treated wastewater were analysed, such as: biological oxygen demand ( BOD 5), chemical oxygen demand ( COD Cr), total suspended solids ( TSS), total nitrogen (N tot), total phosphorus (P tot). The preliminary analysis of pollution indicators series included the: calculation of descriptive statistics and assessment of biological degradability of wastewater. The consistency of the theoretical distributions with the empirical ones was assessed using Anderson–Darling statistics. The best-fitting statistical distributions were selected using the percent bias criterion. Based on the calculations performed, it was found that the analysed indicators of pollution in treated wastewater were characterised by an average variability of composition for BOD 5, COD Cr and TSS, and a high variability of composition for N tot and P tot. The best fitted distribution was log-normal for BOD 5, TSS, N tot and P tot and general extreme values for COD Cr. The simulation carried out using the Monte-Carlo method confirmed that there may be problems associated with the reduction of nutrients (N tot and P tot) the analysed wastewater treatment plant. Results of values obtained of the risk values of negative control of wastewater treatment plant operation for biogenic compounds, different from 1, indicate that the number of exceedances at the outflow may be higher than the acceptable one.

About the informatization object functioning stability assessment in conditions of computer attacks at exponential distribution law of time before the enemy’s impact

Objective. The aim of the study is to develop a mathematical model for assessing the stability of the functioning of an informatization object (IO) in the conditions of computer attacks (CA), assuming that the laws of distribution of random variables before the enemy's impact and the recovery time are distributed exponentially.Method. Application of the method of discrete Markov processes to solve the problem of assessing the stability of IO, which differs from the known approaches in that to describe the state of IO, the concept of "non-returnable state" is introduced, into which the system can move due to the exhaustion of the resource allocated to restore its readiness after a successful CA.Result. As a result of the research, a mathematical model has been developed that allows us to build the stability function of the IO taking into account the intensity of the CA flow and the intensity of restoring its operability, taking into account the limitations on the allocated resource.Conclusion. The application of the method makes it possible to quantify the stability of the functioning of the IO by constructing the survivability function of the IO for conditions under which the flows of attacks and recoveries cannot be assumed stationary and ergodic, and there is also no representative statistics for calculating asymptotic estimates of stability. A quantitative assessment of the stability of the IO for the conditions of the CA is in demand by information security management bodies when making decisions to ensure the protection of information, as well as when justifying the requirements for the system to restore operability.

Effect of Uncertainties in Design Variables on the Hysteresis Response of 2D Steel Moment-Resisting Frames

Reliability theory considers the effect of uncertainty in the modeling and design of structures. Uncertainty in variables such as the material properties, geometric dimensions, and external loads during the analysis of steel moment-resisting frames can have significant effects on structural safety. This study evaluated the safety probability of two two-dimentional steel moment-resisting frames and examined the effect of uncertainly on each design parameter on the maximum frame drift using sensitivity analysis. It also evaluated the robustness index of energy dissipated in the stories of the frames. The uncertainties were modeled with normal probabilistic distribution of random variables. The Monte Carlo simulation method and the maximum story drift constraint were used to calculate the probability of safety and define the limit state function, respectively. Nonlinear time-history analysis was performed on the frames under seven ground motion records having an exceedance probability of 10% in 50 years. The modified Ibarra-Krawinkler behavioral model has been used to define the potential plastic hinges of the frame. The results indicated that the values of the probability of safety may be appropriate for evaluation of the 3-story frame, but be undesirable for a design based on reliability assessment. Sensitivity analysis showed that the steel modulus of elasticity and yield stress values having the greatest effects, whereas the web thickness of the sections and gravity loads applied to the stories had the least effects on the maximum drift of the structure. Using the probability of safety and robustness index, this study has produced useful information for the seismic rehabilitation of structures relating to design variable uncertainty on the hysteresis response for steel moment-resisting frames.

This is the moment for probabilistic loops

We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.