Distribution Of Sum Research Articles

Modelling of Major Flood Arrivals on Chinese Rivers by Switch-time Processes

Nowadays, there is a considerable flood risk in China is. A brief description of major floods on Yangtze River and Huang He River is given. Both big Chinese rivers have long records of floods with severe life and property damages. Quantification of the stochastic behaviour of the largest floods is a key task in the risk assessment and mitigation. An exponential distribution of the time intervals between consecutive floods is assumed in classical study of inter-arrival times of floods. An approach for modelling of flood arrivals on both Chinese rivers by switch-time (ST) processes is proposed. These ST distributions can be considered as distributions of sums of random number exponentially distributed random variables. The proposed model specifies explicitly times of occurrence not only of floods but also of higher risk of potential floods. This approach could be useful for making prognoses of floods and for analysing changes in hydrologic behaviour of rivers.

Stochastic compactness of distributions of sums of independent random variables with finite variances∗

We consider sums of independent random variables within the scheme of series. We focus on the case where every summand has a zero mean and finite variance and sums have unit variances. We obtain a criterion of stochastic compactness (defined by W. Feller) for sequences of distributions of such sums. The condition of uniform asymptotic negligibility of summands is not supposed.

The influence of P/As substitution in the melilite-like Na2Co(P2-xAsx)O7 (x = 0.40 and 0.93) solid solutions.

To investigate the influence of P/As substitution on structures and electrical properties, e.g. the effect on material densities, two new solid P/As-doped solutions, Na2CoP1.60As0.40O7 (disodium cobalt diphosphorus arsenic heptaoxide) and Na2CoP1.07As0.93O7 (disodium cobalt phosphorus arsenic heptaoxide), with melilite-like structures have been synthesized by solid-state reactions. Their unit-cell parameters are in agreement with Vegard's law. The obtained structural models were investigated by the bond valence sum (BVS) and charge distribution (CHARDI) validation tools and, for the latter, the structures are described as being built on anion-centred polyhedra. The frameworks can be described as layered and formed by {[Co(P,As)2O7]2-}∞ slabs, with alkali cations sandwiched between the layers and with the interlayer spaces increased due to P/As substitution. The BVS model was extended to a preliminary simulation of the sodium conduction properties in the studied structural type and suggests that the most probable sodium conduction pathways are bidimensional, at the (002) planes.

Gaussian distribution of short sums of trace functions over finite fields

AbstractWe show that under certain general conditions, short sums of ℓ-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalising results of Erdős–Davenport, Mak–Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.

Exponential–Uniform Distribution

In this paper, we start with a distribution of a random sum of N independent exponential random variables when the random number N has a discrete uniform distribution with positive integer parameter k and show that the probability density function (pdf) and cumulative distribution function (cdf) of this new distribution remain pdf and cdf for any positive real number k. This distribution is proportional to the upper tail probability of the gamma distribution. The hazard rate function of the new model can be constant, decreasing, and increasing depending on its parameters. The statistical properties of the distribution are studied. Estimation of the unknown parameters are discussed by the moments and maximum likelihood methods. The usefulness of the distribution is illustrated by two real datasets.

Climate Policy in the Trump Era: Carbon Tax Rising?

Climate policy in the United States always seems to face strong political headwinds. It is not so much that voters dismiss the threat of climate change, or that they believe climate change is a hoax, but coming up with a fair and effective policy has always seemed so daunting. This Article argues that the simplest answer is not, contrary to initial appearances, daunting at all. The most effective and most efficient climate policy at the federal, state, and local level is a carbon tax. Carbon taxation has always seemed politically implausible, due in large part to an inherent public aversion to taxes, but also due to a vicious campaign of misinformation, paralleling the campaign against climate change itself. An important way of countering this bias and this campaign is to emphasize the revenue side of carbon taxation. Without a robust discussion of the uses of carbon tax revenues, the only salient part of a carbon tax to voters is the all-too-apparent cost. There is no sense of the benefits of revenues. This Article fills this gap by presenting a menu of revenue options, along with a discussion of the macroeconomic and the distributional consequences of the different options. This analysis is an input into a political process, which must ultimately decide on the objective, but only with the guidance of some quantitative analysis. This article closes with an argument for a lump sum distribution approach, in which carbon tax proceeds are returned directly back to carbon taxpayers on a largely per-capita basis. This approach largely insulates the poorest two-thirds from increases in energy and other goods, and may not be macroeconomically inferior to other options, as economists have historically assumed.

Exploring the Effects of Sampling Locations for Calibrating the Huff Model Using Mobile Phone Location Data

The introduction of the Huff model is of critical significance in many fields, including urban transport, optimal location planning, economics and business analysis. Moreover, parameters calibration is a crucial procedure before using the model. Previous studies have paid much attention to calibrating the spatial interaction model for human mobility research. However, are whole sampling locations always the better solution for model calibration? We use active tracking data of over 16 million cell phones in Shenzhen, a metropolitan city in China, to evaluate the calibration accuracy of Huff model. Specifically, we choose five business areas in this city as destinations and then randomly select a fixed number of cell phone towers to calibrate the parameters in this spatial interaction model. We vary the selected number of cell phone towers by multipliers of 30 until we reach the total number of towers with flows to the five destinations. We apply the least square methods for model calibration. The distribution of the final sum of squared error between the observed flows and the estimated flows indicates that whole sampling locations are not always better for the outcomes of this spatial interaction model. Instead, fewer sampling locations with higher volume of trips could improve the calibration results. Finally, we discuss implications of this finding and suggest an approach to address the high-accuracy model calibration solution.

A tight approximate analytical framework for performance analysis of equal gain combining receiver over independent Weibull fading channels

In this paper, a method for approximating the probability distribution of sum of independent and identical Weibull random variables is adopted to analyze the performance of equal gain combiner (EGC) receiver over non-identical Weibull fading channel (WFC). Our main result is to derive a generalized expression of the probability density function (PDF) of the signal-to-noise ratio (SNR) at the EGC output in the case of non-identical WFC. Based on this PDF, accurate approximation of significant performance criteria, such as outage probability (OP), the amount of fading (AoF), and average symbol/bit error probability (ASEP/ABEP), are derived. In addition, we derived the analytical expressions for channel capacities under various adaptation policies such as optimal rate adaptation (ORA), optimal simultaneous power and rate adaptation (OPRA), channel inversion with fixed rate (CIFR), and truncated channel inversion with fixed rate (TCIFR). The proposed mathematical analysis is complemented by several numerical results and validated using Monte Carlo simulation method.

Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions∗

Bounds of the accuracy of the normal approximation to the distribution of a sum of independent random variables are improved under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. These results are extended to Poisson-binomial, binomial and Poisson random sums. Under the same conditions, bounds are obtained for the accuracy of the approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, these bounds are constructed for the accuracy of approximation of the distributions of geometric, negative binomial and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma and Student distributions, respectively. All absolute constants are written out explicitly.

The spatial distribution of rock landforms in the Pohořská Mountains (Pohořská hornatina), Czech Republic

Geomorphological mapping with an emphasis on rock landforms was carried out in the Pohořská Mountains (Cz. Pohořská hornatina) and the positional data acquired were further processed using statistical and cartographical methods. The spatial distribution of rock landforms was investigated in relation to lithology, slope, orientation, and elevation based on an analysis using ArcGIS 9.1. The spatial distribution of rock landforms was primarily determined by the index of distribution Wij = Xi / Yj, where Xi is the percentage representation of landforms in the appropriate category and Yj is the percentage quotient of this category in the entire area studied, and wassecondarily determined according to the sum (sum distribution) of the arithmetic mean and the average deviation.

СТАТИСТИЧЕСКАЯ СТРУКТУРА И ТЕНДЕНЦИИ ИЗМЕНЕНИЯ СЕЗОННЫХ СУММ ОСАДКОВ НА ВОСТОКЕ КАЗАХСТАНА В 1970– 2015 ГГ.

The character of tendencies of change of the average seasonal sums of rainfall in the extensive territory of Kazakhstan can be various and it is essential to influence and even to change the republic structure of sown areas. The main objective of the study is to evaluate and compare statistical characteristics, as well as trends in the changes in average seasonal precipitation over the past decades. The average seasonal values of precipitation sums for spring, first and second half of summer were calculated for six stations in East Kazakhstan, the variability characteristics – mean square deviations (CAO) and coefficients of variation were calculated and analyzed. The characteristics of the normal distribution of the average seasonal sums of precipitation (asymmetry and kurtosis coefficients) have been estimated, histograms of precipitation sums for the above seasons at the Ayaguz, Bakhty, Katon-Karagai, Pavlodar,Semey, Ust-Kamenogorsk and averaged over the eastern Kazakhstan seasonal precipitation. The trendsof these amounts change for each of the above stations and for precipitation averaged over the territoryof East Kazakhstan are determined. Trends in seasonal precipitation in spring, in the first and second halfof summer for individual stations, indicate a significant and insignificant increase in seasonal precipitation,and averaged over the territory of seasonal amounts – for significant growth in recent decades inall the considered seasons of the warm period of the year. This is also indicated by the annual amount ofprecipitation, which increased during the period under study.

Some Remarks on the Asymptotic Behavior of the Empirical Availability Function

We consider the asymptotic behavior of the empirical availability function, which is an important reliability characteristic of technical, communication, information, or transport systems. Our main focus is the case of violation of the classical assumptions of homogeneity of failure flow or the existence of the expectation of failure-free performance or repair duration. Both the classical situation dealing with samples with nonrandom size and the situation where the number of available observations is unknown beforehand, that is, the sample size is random, are considered. In a special situation where the sample size has a negative binomial distribution, an analogue of the law of large numbers is proved for random sums of not necessarily identically distributed random variables describing conditions for the convergence of the distributions of negative binomial random sums to generalized gamma distributions. Thus, a simple limit scheme is proposed, within which generalized gamma distributions turn out to be...

A Geometric Derivation of the Irwin-Hall Distribution

The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

Optimal Distribution of the Given Sum of Node Voltages in Power Systems Electric Networks

The evidence is given for the matrix expression for optimization of the given node voltages sum distribution between the electric network nodes with one and many generation nodes. The regime structures of electric networks, which provide the minimum power losses during the given node voltages sum distribution between the electric network nodes with one and many generation nodes, were proven. The regime structure of the optimal electric network with one generation node is represented as the related population of the initial network arms, involved in the nodal loads power supply. Generally, the regime structures of the optimal electric network with one and many generation nodes are the same as the initial networks structures. In this case, the regime structures are characterized by different nodes voltages. The currents of load nodes are equal and amount to the generation current of the zero balancing point.

MIMO Systems With Restricted Pre/Post-Coding—Capacity Analysis Based on Coupled Doubly Correlated Wishart Matrices

Many practical communication systems have some form of restricted precoding or postcoding, such as antenna selection, selection combining, beam selection, and limited feedback precoding, to name a few. The capacity analysis of such systems is, in general, difficult and previous works in the literature provide results only for certain simplified cases. This paper derives a novel approach to analyze the capacity for such systems under a very generic setting. The results are based on asymptotic closed-form expressions for the second-order statistics and joint distributions of eigenvalues for a set of coupled, doubly correlated Wishart matrices. A tight approximation to the joint distribution of the eigenvalues in the non-asymptotic regime is also proposed. These results are then used to show that the system capacity can be approximated as the largest element of a correlated Gaussian vector. Showing that this is equivalent to the problem of finding the distribution of sum of lognormals, we propose a novel approach to characterize its distribution. As an application, the capacity for an antenna selection system and a limited feedback precoding system is compared with their respective approximations. This paper also demonstrates how the results can be used to design the precoding codebook in limited feedback systems.

Kolmogorov goodness-of-fit test for S-symmetric distributions in climate and weather modeling

Statistical data treatment is an essential part of climate and weather modeling. The Kolmogorov goodness-of-fit test is a widely applicable statistical method to determine the cumulative distribution function of a continuous random variable, e.g., a precipitation level, wind force, etc. In this paper, the authors consider a problem of goodness-of-fit testing involving additional information about S-symmetry of the cumulative distribution function and its influence on the Kolmogorov statistic distributions. A definition of S-symmetry is given; it is a generalized classical definition of distribution symmetry. It is proved that any continuous increasing cumulative distribution function is S-symmetric. A uniform distribution is considered as an example of an S-symmetric distribution. A modification of the Kolmogorov statistic using additional information about the new type of symmetry is proposed. The exact and asymptotic distributions under the null and the alternative hypothesis of the modified statistics are described. The authors also provide an example which proves that the modified test is more powerful than the non-modified one. The new test is used to check the hypothesis of a uniform distribution of the average sum of precipitation.

Discriminating membrane proteins using the joint distribution of length sums of success and failure runs

Discriminating integral membrane proteins from water-soluble ones, has been over the past decades an important goal for computational molecular biology. A major drawback of methods appeared in the literature, is that most of the authors tried to solve the problem using machine learning techniques. Specifically, most of the proposed methods require an appropriate dataset for training, and consequently the results depend heavily on the suitability of the dataset, itself. Motivated by these facts, in this paper we develop a formal discrimination procedure that is based on appropriate theoretical observations on the sequence of hydrophobic and polar residues along the protein sequence and on the exact distribution of a two dimensional runs-related statistic defined on the same sequence. Specifically, for setting up our discrimination procedure, we study thoroughly the exact distribution of a bivariate random variable, which accumulates the exact lengths of both success and failure runs of at least a specific length in a sequence of Bernoulli trials. To investigate the properties of this bivariate random variable, we use the Markov chain embedding technique. Finally, we apply the new procedure to a well-defined dataset of proteins.

A Note on Mixture Representations for the Linnik and Mittag-Leffler Distributions and Their Applications*

We present some product representations for random variables with the Linnik, Mittag-Leffler, and Weibull distributions and establish the relationship between the mixing distributions in these representations. The main result is the representation of the Linnik distribution as a normal scale mixture with the Mittag-Leffler mixing distribution. As a corollary, we obtain the known representation of the Linnik distribution as a scale mixture of Laplace distributions. Another corollary of the main representation is the theorem establishing that the distributions of random sums of independent identically distributed random variables with finite variances converge to the Linnik distribution under an appropriate normalization if and only if the distribution of the random number of summands under the same normalization converges to the Mittag-Leffler distribution.

Product Representations for Random Variables with Weibull Distributions and Their Applications*

In this paper, product representations are obtained for random variables with theWeibull distribution in terms of random variables with normal, exponential and stable distributions yielding scale mixture representations for the corresponding distributions. Main attention is paid to the case where the shape parameter γ of theWeibull distribution belongs to the interval (0, 1]. The case of small values of γ is of special interest, since the Weibull distributions with such parameters occupy an intermediate position between distributions with exponentially decreasing tails (e.g., exponential and gamma-distributions) and heavy-tailed distributions with Zipf–Pareto power-type decrease of tails. As a by-product result of the representation of the Weibull distribution with γ ∈ (0, 1) in the form of a mixed exponential distribution, the explicit representation of the moments of symmetric or one-sided strictly stable distributions are obtained. It is demonstrated that if γ ∈ (0, 1], then the Weibull distribution is a mixed half-normal law, and hence, it can be limiting for maximal random sums of independent random variables with finite variances. It is also demonstrated that the symmetric two-sided Weibull distribution with γ ∈ (0, 1] is a scale mixture of normal laws. Necessary and sufficient conditions are proved for the convergence of the distributions of extremal random sums of independent random variables with finite variances and of the distributions of the absolute values of these random sums to the Weibull distribution as well as of those of random sums themselves to the symmetric two-sided Weibull distribution. These results can serve as theoretical grounds for the application of the Weibull distribution as an asymptotic approximation for statistical regularities observed in the scheme of stopped random walks used, say, to describe the evolution of stock prices and financial indexes. Also, necessary and sufficient conditions are proved for the convergence of the distributions of more general regular statistics constructed from samples with random sizes to the symmetric two-sided Weibull distribution.

The New Mortality Tables, Life Expectancy and Pension Plans

The Internal Revenue Service will adopt new regulations replacing outdated mortality tables RP-2000 and improvement factor AA with RP-2014 and MP-2016 (RP for “retirement plan” and MP for “mortality projection”) for plan years beginning on or after January 1, 2018. The new tables will affect plan valuation (that drive employer contributions), Pension Benefit Guaranty Corporation’s variable rate premiums and the calculation of lump sum distributions. They will substantially increase the cost of sponsoring a traditional defined benefit pension plan. This article explains the workings and function of mortality tables and life expectancy, traces the development of the new tables by the Society of Actuaries and their adoption by the IRS, assesses their likely impact and examines probable responses by plan sponsors.