Showing papers on "Probability density function published in 1985"

Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities: The Inverse Problem

Abstract: The inverse problem is defined here as follows: determine the transmissivity at varius points, given the shape and boundary of the aquifer and recharge intensity and given a set of measured log-transmissivity Y and head H values at a few points. The log-transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function (pdf). The solution of the inverse problem is the conditional normal pdf of Y, conditioned on measured H and Y, which is expressed in terms of the unconditional joint pdf of Y and H. The problem is reduced to determining the unconditional head-log-transmissivity covariance and head variogram for a selected Y covariance which depends on a few unknown parameters. This is achieved by solving a first-order approximation of the flow equations. The method is illustrated for an exponential Y covariance, and the effect of head and transmissivity measurements upon the reduction of uncertainty of Y is investigated systematically. It is shown that measurement of H has a lesser impact than those of Y, but a judicious combination may lead to significant reduction of the predicted variance of Y. Possible applications to real aquifers are outlined.

Bounding the Project Completion Time Distribution in PERT Networks

Abstract: We consider the PERT model of a project composed of activities whose durations are random variables with known distributions. For the situations in which the activity durations are completely independent, we present a new method for obtaining a probability distribution function that bounds the exact probability distribution of the project completion time from below. The bounding distribution can be used to obtain an upper bound on the mean completion time of the project. We also prove and illustrate that this bounding distribution is better tighter than any of the existing lower bounds, implying that the corresponding upper bound on the mean completion time is tighter than any of the existing upper bounds.

I–K distribution as a universal propagation model of laser beams in atmospheric turbulence

Abstract: A new propagation model is developed for the intensity fluctuations of a laser beam propagating through extended clear-air turbulence. The field of the optical wave is modeled as the sum of a coherent (deterministic) component and a random component, the intensity of which is assumed governed by the generalized n distribution of Nakagami. We further assume that the statistics are inherently nonstationary by treating the average intensity of the random portion of the field as a fluctuating quantity. The resulting unconditional I–K distribution for the intensity fluctuations is a generalized form of the K distribution that is applicable to all conditions of atmospheric turbulence for which data have been obtained, including weak turbulence for which the K distribution is not theoretically applicable.

Unit hydrograph approximations assuming linear flow through topologically random channel networks

Abstract: The instantaneous unit Hydrograph (IUH) of a drainage basin is derived in terms of fundamental basin characteristics (Z, α, β), where α parameterizes the link (channel segment) length distribution, and β is a vector of hydraulic parameters, Z is one of three basin topological properties, N, (N, D), or (N, M), where N is magnitude (number of first-order streams), D is diameter (mainstream length), and M is order. The IUH is derived based on assumptions that the links are independent and identically distributed random variables and that the network is a member of a topologically random population. Linear routing schemes, including translation, diffusion, and general linear routing are used, and constant drainage density is assumed. By using (N, α, β) as the fundamental basin characteristics, asymptotic (for large N) considerations lead to a Weibull probability density function for the IUH, with time to peak given by tp = (2N)½ α*/β* where α* is mean link length, and β* is a scalar hydraulic parameter (usually average celerity). This asymptotic IUH is identical for all linear routing schemes.

Nonparametric Kernel Estimation of Flood Frequencies

Abstract: A currently used approach to flood frequency analysis is based on the concept of parametric statistical inference. In this analysis the assumption is made that the distribution function describing flood data is known, for example, a log-Pearson type III distribution. However, such an assumption is not always justified and often leads to other difficulties; it could also result in considerable variability in the estimation of design floods. A new method is developed in this article based on the nonparametric procedure for estimating probability distribution function. The results indicate that design floods computed from the different assumed distribution and from the nonparametric method provide comparable results. However, the nonparametric method is a viable alternative with the advantage of not requiring a distributional assumption, and has the ability of estimating multimodal distributions.

The Role of AM-to-PM Conversion in Memoryless Nonlinear Systems

Abstract: A generalized proof is presented that AM-to-PM conversion can only degrade, never improve, the intermodulation-noise performance of memoryless nonlinear systems with random input signals having even probability density functions, and a measure of degradation is defined. It is also shown for such signals that AM-to-PM conversion causes a deterministic constant phase shift to be added to the argument of the signal component at the output but has no other effect on its phase. This class of inputs includes one or the sum of several PSK signals, as well as large ensembles that can be modeled as Gaussian noise. The latter are dealt with by using Bussgang's theorem on input-output cross correlation. In the proof, Bussgang's theorem is extended to the complex case, to include phase as well as amplitude nonlinearities, yielding a complex version of the theorem. For Gaussian inputs it is shown that the undistorted signal and the intermodulation noise at the output of such systems are uncorrelated.

Measurement of the Wigner Function

Abstract: An operational procedure is given for determining experimentally, in principle at least, the Wigner function for an ensemble of particles. This manner of "measuring" a quantum state, whether pure or mixed, via its Wigner function, seems the simplest possible, and closely parallels the method one might use in classical mechanics to determine a (true) phase-space probability density.

A population-density approach to regional spatial structure

Abstract: The concept of the population-density function, which is usually applied within the context of an urban area, is extended to the scale of a metropolitan-area-based region or city region. A form for such a regional density function is proposed, and this is examined for selected regions of the UK and North America. It is demonstrated how such a region-wide density function may be related to other density functions which have been used to characterise the structures of the metropolitan and non-metropolitan parts of the region. Consideration is also given to problems associated with constructing and interpreting the regional density function and to its underlying theoretical basis. Finally, the application of the density-function approach in the analysis of regional structure is briefly discussed.

The Statistical Properties of Partially Polarized Light

Abstract: The focus which the end-point of the electric vector describes at a fixed point in space is an ellipse characterized by three parameters: the sum of the squares of the amplitudes of the two electric vector components, the angle which the major axis makes with the reference coordinate system, and the ratio of the minor to the major axis The joint and marginal probability densities of these random variables are determined as a function of the degree of polarization Results are obtained numerically for the mean, variance and marginal probabilities of these random variables as a function of the degree of polarization

Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous

Abstract: Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sampleX 1, …,X n, were obtained by Rosenblatt [4] and Epanechnikov [1] for the case when the densityf and its derivativef′ are continuous. They found, under certain additional regularity conditions, that the optimal choiceh n0 for the scale factorh n=Kn−α is given byh n0=K0n−1/5 withK 0 depending onf and the kernel; they also showed that MISE(h n0)=O(n−4/5) and Epanechnikov [1] found the optimal kernel. In this paper we investigate the robustness of these results to departures from the assumptions concerning the smoothness of the density function. In particular it is shown, under certain regularity conditions, that whenf is continuous but its derivativef′ is not, the optimal value of α in the scale factor becomes 1/4 and MISE(h n0)=O(n−3/4); for the case whenf is not continuous the optimal value of α becomes 1/2 and MISE(h n0)=O(n−1/2). For this last case the optimal kernel is shown to be the double exponential density.

On the Probability Density Function of the Squared Envelope of a Sum of Random Phase Vectors

Abstract: A recursive solution is presented for the probability density function of the sum of N independent, random phase vectors. The recursion parameter is N , the number of vectors in the sum. This approach allows one to rapidly compute a complete set of these density functions for values of N = 2, 3, ..., N_{max} , where N max typically corresponds to the total number of system users in a multiuser FHMA/MFSK application, or one plus the total number of jamming tones in an FH/MFSK spread-spectrum application. Such evaluations are necessary for exact calculation of the average error probability performance of such systems.

A Probabilistic Insulation Life Model for Combined Thermal-Electrical Srresses

Abstract: The Weibull distribution is widely used in statistical problems related to aging of solid insulating materials subjected to electrical stress. The main object of this paper is to explain the Weibull probability function in such a way that it can be applied to the statistical analysis of the risk of failure for solid insulating materials or structures subjected to single or combined (in particular thermal-electrical) stress situations. For this purpose, appropriate expressions for the scale and shape parameters of the two-parameter Weibull function are proposed, starting from a model for combined-life, based on the inverse power model for electrical life and the Arrhenius relationship for thermal life. The agreement of the statistical model thus obtained has been verified by means of experimental tests carried out on Low-Density Polyethylene.

Joint statistics between temperature and its dissipation in a turbulent jet

Abstract: The joint probability density function of the temperature θ and its instantaneous dissipation χ is important for the development of turbulent reacting flow models. Measurements are presented, in the nearly self‐preserving region of a turbulent plane jet, of the joint statistics of θ and an approximation χ1 to χ. Properties of the three, separately obtained, dissipation components are first compared to justify the choice of χ1. The quantity chosen is an appropriately weighted linear combination of the squared values of the temporal derivative, shown to be an adequate substitution for the derivative in the streamwise direction, and the derivative, in either the lateral or spanwise directions, of temperature. The correlation between θ and χ1 is weak, receiving contributions primarily at small frequencies. The assumption of independence between θ and χ1, allowing the joint probability density function of these two quantities to be written as a product of the individual probability density functions, improves with distance from the jet centerline.

Global asymptotic stability of the size distribution in probabilistic models of the cell cycle.

Abstract: Probabilistic models of the cell cycle maintain that cell generation time is a random variable given by some distribution function, and that the probability of cell division per unit time is a function only of cell age (and not, for instance, of cell size). Given the probability density, f(t), for time spent in the random compartment of the cell cycle, we derive a recursion relation for φ n(x), the probability density for cell size at birth in a sample of cells in generation n. For the case of exponential growth of cells, the recursion relation has no steady-state solution. For the case of linear cell growth, we show that there exists a unique, globally asymptotically stable, steady-state birth size distribution, φ *(x). For the special case of the transition probability model, we display φ *(x) explicitly.

The Existence of the First Negative Moment

Abstract: The question of the existence of negative moments of a continuous probability density function is explored. A sufficient condition for the existence of the first negative moment is given. The condition is easy to verify, as it involves limits rather than integrals. An example is given, however, that shows that this simple condition is not necessary for the existence of the first negative moment. The delicacy of the characterization of existence is explored further with some results concerning the existence of moments surrounding the first negative moment.

The form of the regional density function

Abstract: Parr J. B. (1985) The form of the regional density function, Reg. Studies 19, 535–546. The concept of the population density function is considered in the context of a metropolitan-area-based region. The observed characteristics of such a density function for the different parts of a region are outlined, and three mathematical functions are examined in terms of their ability to reflect these characteristics. Two of these functions, which have been widely employed in a metropolitan-area context, are shown to be inappropriate at the regional scale. A third function, the lognormal function, appears to be capable of reproducing these characteristics. Various aspects of this function are discussed, including its use in the analysis of spatial-structure development in highly urbanized regions.

Performance evaluation of a flexible manufacturing cell with random multiproduct feedback flow

Abstract: This paper presents a predictive model for describing the productive capacity of special multiproduct manufacturing cells with stochastic activity times and random feedback flow. Two cases are considered, single-product type batch production and multiproduct type interleaved production. The major performance measures which are evaluated are the total batch processing time, number of product recycles and the distribution of interdeparture times for each product type. Expressions are given for the distribution functions, means and variances of these quantities, for any distribution of manufacturing time. Operational analysis of a flexible manufacturing cell is discussed with an illustration of the Oscillatory phenomenon of the probability density function of its total batch time.

Probability densities for the displacement of random walks on percolation clusters

Abstract: The probability density of the displacement or end-to-end distance of a random walk on the incipient infinite percolation cluster in d=2 dimensions is studied by an exact enumeration method The numerical data suggest specific forms for the probability density both in the chemical distance variable l and the geometric distance r

Probability current versus charge current of a relativistic particle

Abstract: The probability density of detecting a relativistic particle has been found by Newton and Wigner (1949) for a point in space. This probability density must, however, be the zero component of the same vector field. This (unique) vector field-the probability current-is constructed and discussed.

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Abstract: Given a random sample of sizen from a densityf 0 on the real line satisfying certain regularity conditions, we propose a nonparametric estimator forψ 0=−f 0 ′ /f0. The estimate is the minimizer of a quadratic functional of the formλJ(ψ)+∫[ψ 2−2ψ′]dFn where λ>0 is a smoothing parameter,J(·) is a roughness penalty, andF n is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which is related to spline functions. A more complete study of the caseJ(ψ)=∫[d 2ψ/dx2]2 is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness limit (λ→∞). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem is simple in comparison with the analogous penalized likelihood estimators.

The Sequential Probability Ratio Test

Abstract: We begin by recalling the Neyman-Pearson Lemma for testing a simple hypothesis against a simple alternative. Let x denote a (discrete or continuous) random variable (or vector) with probability density function f.

Steady-State Size Distributions in Probabilistic Models of the Cell Division Cycle

Abstract: A model for the steady-state size distribution in an exponentially growing population of single cells is derived and studied. The model incorporates a general growth law for individual cells and a probability density for interdivision times. A uniqueness theorem is proved, and it is shown that no solution exists when individual cells grow exponentially. For linear growth, infinite series solutions are found in two specific cases. Statistical data are obtained for these solutions, and comparisons are made with the results of some numerical simulations and some experiments.