Moments Of Probability Distribution Research Articles

Formulas of Absolute Moments

The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat Hsu’s (J. Chinese Math. Soc. N.S. 1, 257–280, 1951) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace–Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely determined by a proper subset of the translated fractional moments. This improves significantly Hall’s (Z. Wahrsch. Verw. Gebiete 62, 355–359, 1983) result for distributions on the right half-line.

Идентификация псевдослучайных последовательностей по моментам вероятностных распределений

Идентификация псевдослучайных последовательностей по моментам вероятностных распределений

Recurrent Algorithms of Structural Classification Analysis for Complex Organized Information

For the structural classification analysis of complex organized information, we propose to use recurrent algorithms of stochastic approximation type. We introduce classification quality functionals that depend on non-normalized and zero moments of probability distribution functions for the probability of sample objects appearing in the classes, as well as the type of optimal classification. We propose a new classification algorithm for this type of classification quality criteria and prove a theorem about its convergence that ensures the stationary value of the corresponding functional. We show that the proposed algorithm can be used to solve a wide class of problems in structural classification analysis.

On generating functions of Hausdorff moment sequences

The class of generating functions for completely monotone sequences (moments of finite positive measures on [ 0 , 1 ] [0,1] ) has an elegant characterization as the class of Pick functions analytic and positive on ( − ∞ , 1 ) (-\infty ,1) . We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [ 0 , 1 ] [0,1] . Also we provide a simple analytic proof that for any real p p and r r with p > 0 p>0 , the Fuss-Catalan or Raney numbers r p n + r ( p n + r n ) \frac {r}{pn+r}\binom {pn+r}{n} , n = 0 , 1 , … n=0,1,\ldots , are the moments of a probability distribution on some interval [ 0 , τ ] [0,\tau ] if and only if p ≥ 1 p\ge 1 and p ≥ r ≥ 0 p\ge r\ge 0 . The same statement holds for the binomial coefficients ( p n + r − 1 n ) \binom {pn+r-1}n , n = 0 , 1 , … n=0,1,\ldots \, .

Price-based vs. quantity-based environmental regulation under Knightian uncertainty: An info-gap robust satisficing perspective

Conventional wisdom among environmental economists is that the relative slopes of the marginal social benefit and marginal social cost functions determine whether a price-based or quantity-based environmental regulation leads to higher expected social welfare. We revisit the choice between price-based vs. quantity-based environmental regulation under Knightian uncertainty; that is, when uncertainty cannot be modeled with known moments of probability distributions. Under these circumstances, the policy objective cannot be to maximize the expected net benefits of emissions control. Instead, we evaluate an emissions tax and an aggregate abatement standard in terms of maximizing the range of uncertainty under which the welfare loss from error in the estimates of the marginal benefits and costs of emissions control can be limited. The main result of our work is that the same criterion involving the relative slopes of the marginal benefit and cost functions determines whether price-based or quantity-based control is more robust to unstructured uncertainty. Hence, not only does the relative slopes criterion lead to the policy that maximizes the expected net benefits of control under structured uncertainty, it also leads to the policy that maximizes robustness to unstructured uncertainty.

Polarization properties of fluctuations of the field created by randomly located dipoles

The vector properties of the strength of a field created by an ensemble of N parallel dipoles located, on average, uniformly are considered. For N → ∞, the problem is reduced to the Poisson problem. The direction of the dipoles specifies the symmetry axis of fluctuations. The moments of probability distributions for the Cartesian components of the strength are calculated. The anisotropy of the fluctuations is approximately 15–20%. Under conditions that are of interest for spectroscopic applications, the N-particle probability distribution contains a weak broad negative background, the shape of which replicates the one-particle distribution. The analogy between the problem considered and the theory of spectral line broadening in the impact approximation, in particular, the Dicke narrowing of the Doppler profile, is analyzed. Conditions of applicability of the theory developed are discussed.

Unsteady wind loading on a wall

This paper presents an extensive analysis of unsteady wind loading data on a 18 m long andrn2 m high wall in a rural environment, with the wind at a range of angles to the wall normal. The data isrnfirstly analyzed using standard statistical techniques (moments of probability distributions, auto- and cross-correlations,rnauto- and cross-spectra etc.). The analysis is taken further using a variety of less conventionalrnmethods - conditional sampling, proper orthogonal decomposition and wavelet analysis. It is shown that,rneven though the geometry is simple, the nature of the unsteady flow is surprisingly complex. The fluctuatingrnpressures on the front face of the wall are to a great extent caused by the turbulent fluctuations in thernupstream flow, and reflect the oncoming flow structures. The results further suggest that there are distinctrnstructures in the oncoming flow with a variety of scales, and that the second order quasi-steady approachrncan predict the pressure fluctuations quite well. The fluctuating pressures on the rear face are alsorninfluenced by the fluctuations in the oncoming turbulence, but also by unsteady fluctuations due to wakernunsteadiness. These fluctuations have a greater temporal and spatial coherence than on the front face andrnthe quasi-steady method over-predicts the extent of these fluctuations. Finally the results are used to checkrnsome assumptions made in the current UK wind loading code of practice.