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1
Content available remote New easy to compute formulas for the moments of random variables appearing in the coupon collectorproblem
León-García Amaryani, Pérez Aroldo, Bolívar-Cimé Addy
Probability and Mathematical Statistics
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2023
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Vol. 43, Fasc. 2
155--164
EN
Assuming that there are N types of coupons, where the prob- ability that the ith coupon appears is pi ≥ 0 for i = 1, . . . , N , with [formula], we consider the variable Tk which represents the number of acquisitions needed to obtain k ≤ N different coupons, and the variable Yn which represents the number of different coupons obtained in n acquisitions. In the coupon collector problem it is of interest to obtain the expected value of these random variables, as well as their rth moments. We provide new expressions for the rth moments of Tkand Yn, and we give expressions for their moment generating functions. Unlike known formulas, our formula for the rth moment of Tk is given in terms of recursive expressions and that of Yn is given in terms of finite sums, so that they can be easily implemented computationally. Furthermore, our formulas allow obtaining simplified expressions of the first few moments of the variables.
2
Content available The application of cumulants to flow routing
Romanowicz Renata J., Doroszkiewicz Joanna
Meteorology Hydrology and Water Management. Research and Operational Applications
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2019
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Vol. 7, Iss. 1
15--21
EN
This paper aims to fill a gap between present and past research approaches to modelling flow in open channels. In particular, a history of the analytical solutions of a linearized St. Venant equation is presented. A solution of the linearized St. Venant equation, describing the response of a river channel to a single impulse forcing, the so called Instantaneous Unit Hydrograph (IUH), can be described using cumulants, defined as the moments of a logarithm of a variable. A comparison of analytical and numerical solutions of flood wave propagation under various flow conditions is given. The river reach of Biała Tarnowska is used as an illustration of both approaches. A practical application of simplified solutions to the emulator of a flood wave propagation is suggested showing a link between theory and practice.
3
Content available remote Moments of the weighted Cantor measures
Harding Steven N., Riasanovsky Alexander W. N.
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
256--273
EN
Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying [wzór] where φn : x ↦ (x+n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L2μα[0,1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xmdμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the firstmmoments within uniform error ε in O((loglog(1/ε))·m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2,1/2].
4
Content available remote Application of the Moment Shape Representations to the General Shape Analysis
Frejlichowski D.
Image Processing & Communications
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2016
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Vol. 21, no. 4
31--38
EN
The General Shape Analysis (GSA) is a task similar to the shape recognition and retrieval. However, in GSA an object usually does not belong to a template class, but can only be similar to some of them. Moreover, the number of applied templates is limited. Usually, ten most general shapes are used. Hence, the GSA consists in searching for the most universal information about them. This is useful when some general information has to be concluded, e.g. in coarse classification. In this paper the result of the application of three shape descriptors based on the moment theory to the GSA is presented. For this purpose the Moment Invariants, Contour Sequence Moments, and Zernike Moments were selected.
5
Content available remote On pointwise convergence of nets of Mellin-Kantorovich convolution operators
Bardaro C., Mantellini I.
Commentationes Mathematicae
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2013
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Vol. 53, No. 2
65--77
EN
Here we study pointwise approximation and asymptotic formulae for a class of Mellin-Kantorovich type integral operators, both in linear and nonlinear form.
6
Content available Metoda analizy niepewności oparta na połączeniu zasady maksymalnej entropii i metody oceny punktowej
Zhang X. L., Huang H. Z., Wang Z. L., Xiao N. C., Li Y. F.
Eksploatacja i Niezawodność
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2012
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Vol. 14, No. 2
114-119
PL
Niepewność jest nieodłącznym elementem procesów projektowania produktu. Dlatego też podejmowanie niezawodnych decyzji wymaga analizy niepewności, która uwzględniałaby wszystkie rodzaje niepewności. W praktyce inżynierskiej, z powodu niepełnej wiedzy, wyznaczenie rozkładu niektórych zmiennych projektowych nie jest możliwe. Co więcej, funkcja stanu granicznego jest wysoce nieliniowa, co sprawia, że do poprawnego obliczenia prawdopodobieństwa uszkodzenia potrzebna jest znajomość momentów wyższych rzędów tej funkcji. W niniejszej pracy zaproponowano metodę analizy niepewności łączącą zasadę maksymalnej entropii z metodą bootstrapową. W pierwszej części pracy wykorzystano metodę bootstrapową do obliczenia przedziałów ufności czterech pierwszych momentów dla zmiennych losowych typu mieszanego oraz zmiennych z próby. Następnie, wyznaczono momenty wyższych rzędów funkcji stanu granicznego przy użyciu metody redukcji wymiarów. Po trzecie, w celu obliczenia funkcji gęstości prawdopodobieństwa (PDF) oraz dystrybuanty (CDF) funkcji stanu granicznego, sformułowano model optymalizacji oparty na zasadzie maksymalnej entropii. Proponowana metoda nie wymaga założenia znajomości rozkładów zmiennych losowych ani obliczania wrażliwości dla funkcji stanu granicznego w odniesieniu do najbardziej prawdopodobnego punktu awarii. W końcowej części artykułu porównano na podstawie przykładów numerycznych wyniki otrzymane za pomocą proponowanej metody oraz symulacji Monte Carlo (MCS).
EN
Uncertainty is inevitable in product design processes. Therefore, to make reliable decisions, uncertainty analysis incorporating all kinds of uncertainty is needed. In engineering practice, due to the incomplete knowledge, the distribution of some design variables can not be determined. Furthermore, the performance function is highly nonlinear, therefore, the high order moments of the performance function are needed to calculate the probability of failure accurately. In this paper, an uncertainty analysis method combining the maximum entropy principle and the bootstrapping method is proposed. Firstly, the bootstrapping method is used to calculate the confidence intervals of the first four moments for mixed random variables and sample variables. Secondly, the high order moments of limit state functions are estimated using the reduced dimension method. Thirdly, to calculate the probability density function (PDF) and cumulative distribution function (CDF) of the limit state functions, an optimization model based on the maximum entropy principle is formulated. In the proposed method, the assumptions that the distribution of the random variables are known and the calculation of the sensitivity for limit state function with respect to the Most Probable Point (MPP) are avoided. Finally, comparisons of results from the proposed methods and the MCS method are presented and discussed with numerical examples.
7
Content available remote Improved bounds on bell numbers and on moments of sums of random variables
Berend D., Tassa T.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 2
185--205
EN
We provide bounds for moments of sums of sequences of independent random variables. Concentrating on uniformly bounded nonnegative random variables, we are able to improve upon previous results due to Johnson et al. [10] and Latała [12]. Our basic results provide bounds involving Stirling numbers of the second kind and Bell numbers. By deriving novel effective bounds on Bell numbers and the related Bell function, we are able to translate our moment bounds to explicit ones, which are tighter than previous bounds. The study was motivated by a problem in operation research, in which it was required to estimate the Lp-moments of sums of uniformly bounded non-negative random variables (representing the processing times of jobs that were assigned to some machine) in terms of the expectation of their sum.
8
Content available remote Axial dispersion models and their basic properties
Bartova D., Jakes B., Kukal J.
Archives of Control Sciences
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2009
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Vol. 19, no. 1
5-22
EN
The paper is oriented to summary of important basic relations, which characterize behavior of four axial dispersion models (AEO: axial enforced closed-open model, ACO: axial closed-open model, ACC: axial closed-closed model, AOO: axial open-open model) and three referential models (ideal mixed model, plug flow model, cascade of ideal mixers without back-mixing). Selected basic properties (parametric characteristics) of these models can be used for parameter identification of included hydrodynamic flow structure models. Mathematical description of models including initial and boundary conditions, transfer function, model transient response to Dirac impulse as weighting (impulse) function, model transient response to step function as step response are included in this study. There are also included further characteristics of impulse function: raw moments up to 4th order, variance, variation coefficient, skewness , kurtosis, location and value of mode. Complete set of these characteristics for all studied models is collected (model-by-model) in seven tables. The authors declare several properties of weighting function as key ones: value of 1st raw (dimensional) moment, parametric values and mode properties, related to dependence on Peclet number. The plots of parametric values and mode properties vs. Peclet number are mentioned in the paper for four studied axial dispersion models.
9
Content available remote Korovkin theorem in modular spaces
Bardaro C., Mantellini I.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2007
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Vol. 47, [Z] 2
239-253
EN
In this paper we obtain an extension of the classical Korovkin theorem in abstract modular spaces. Applications to some discrete and integral operators are discussed.
10
Content available remote Analiza dynamiczna prostokątnej płyty sprężystej poddanej działaniu fali uderzeniowej
Paluch M.
Czasopismo Techniczne. Budownictwo
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2006
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R. 103, z. 13-B
107-115
PL
W niniejszym artykule wyprowadzono dla płyty sprężystej równania ruchu, a następnie dla obciążenia dynamicznego, jakim jest fala uderzeniowa, wyznaczono analitycznie tensor uogólnionych sił na jednostkę długości płyty. Rozwiązania podano w formie analitycznej.
EN
This paper deals with the dynamic bending analysis of plates. Plate motion equation, bending moments, twisting moment and the shearing forces were determined.
11
Physics of flood frequency analysis. Part I. Linear convective diffusion wave model
Strupczewski W., Węglarczyk S., Singh V.
Acta Geophysica Polonica
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2002
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Vol. 50, nr 3
433-455
EN
It is hypothesized that the impulse response of a linearized convective diffusion wave (CD) model is a probability distribution suitable for flood frequency analysis. This flood frequency model has two parameters, which are derived using the methods of moments and maximum likelihood. Also derived are errors in quantiles for these methods of parameter estimation. The distribution shows an equivalency of the two estimation methods with respect to the mean value - an important property in the case of unknown true distribution function. As the coefficient of variation tends to zero (with the mean fixed), the distribution tends to a normal one, similar to the lognormal and gamma distributions.
12
Content available remote The generalization of the Kac-Bernstein Theorem
Quine M. P., Seneta E.
Probability and Mathematical Statistics
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1999
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Vol. 19, Fasc. 2
441--452
EN
The Skitovich-Darmois Theorem of the early 1950's establishes the normality of independent X1, X2,…, Xn from the independence of two linear forms in these random variables. Existing proofs generally rely on the theorems of Marcinkiewicz and Cramér, which are based on analytic function theory. We present a self-contained real-variable proof of the essence of this theorem viewed as a generalization of the case n = 2, which is generally called Bernstein's Theorem, and also adapt an early little known argument of Kac to provide a direct simple proof when n = 2. A large bibliography is provided.
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