Probability density function correlation functions

Since the histogram gives a probability density function of the particle position, the correlation in the velocities Vy and V2j in the j-direction causes the change in the shape of the histogram plotted against Vy and V2j, due to the different coefficient y — Pj in... 

The probability density function W(0) at r = 0 and the directional correlation factors for homologous PACA sequences with x = 2 to 7 units are evaluated. The influences of these factors on the cyclization equilibria constants Kx are determined. Agreement of theory with experimental results for x = 3 - 6 is within limits set by uncertainties in the calculations combined with experimental errors, i.e., within about 15 %. 

The averaging operation for the liquid droplet velocity described in the previous section introduces a particle velocity deviation from the mean (or correlated) velocity, noted as m" = Up — ui, and named the random uncorrelated velocity [280]. By definition, the statistical average (based on the particle probability density function) of this uncorrelated velocity is zero < u" >= 0. A conservation equation can be written for the associated kinetic energy 59i =< Up pip > /2 ... 

Equation 6.50 assumes that the asperity width is not correlated with the height that is, the joint probability density function for both height and width is separable. Equation 6.50 should be used in place of (6.41) in the topography polishing model when features have sizes in the filtering regime. 

A set of observed data points is assumed to be available as samples from an unknown probability density function. Density estimation is the construction of an estimate of the density function from the observed data. In parametric approaches, one assumes that the data belong to one of a known family of distributions and the required function parameters are estimated. This approach becomes inadequate when one wants to approximate a multi-model function, or for cases where the process variables exhibit nonlinear correlations [127]. Moreover, for most processes, the underlying distribution of the data is not known and most likely does not follow a particular class of density function. Therefore, one has to estimate the density function using a nonparametric (unstructured) approach. 

Particles, such as molecules, atoms, or ions, and individuals, such as cells or animals, move in space driven by various forces or cues. In particular, particles or individuals can move randomly, undergo velocity jump processes or spatial jump processes [333], The steps of the random walk can be independent or correlated, unbiased or biased. The probability density function (PDF) for the jump length can decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting time between successive jumps can decay rapidly or exhibit a heavy tail. We will discuss these various possibilities in detail in Chap. 3. Below we provide an introduction to three transport processes standard diffusion, tfansport with inertia, and anomalous diffusion. 

For molecular fluids, it is convenient to define different types of distribution functions, correlation functions and related quantities. In particular, in the pair-wise additive theory of homogeneous fluids (see eq. [8.74]), a central role is played by the angular pan-correlation function g(ri2C0i(02) proportional to the probability density of finding two molecules with position rj and rj and orientations (Oi and (Dj (a schematic representation of such function is reported in Figure 8.5). 

The probability density function T(X X) can be any function which satisfies the condition that if T(X X) is not zero T(X X ) is also not zero. Computationally feasible functions are chosen as the probability density function T X X) for the computational convenience. The vector set X, generated with this algorithm has the serial correlation, which means that the sequential position vectors such as X,- and X,+i are correlated. Thus, the error estimation may be modified with the auto-correlation time [20, 21] in the practical QMC calculations. In addition, the modification to the central limit theorem is also proposed in recent QMC study [22]. 

In this case study, the probabilities p(H 0, ai), p(E 9, i) and p(F 0,ai) were estimated separately because the consequences of the three dimensions are considered to occur randomly and independently, with negligible correlation. Consideration was given to the probability density functions of f (H 0, ai), f (E 0, a ) and f (F 9, aj) as representatives of these probabilities in this case study. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

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