Distribution function moments

To write down Eq. (4.53) as well as (4.51), we used, after Ref. 54, the distribution function moments presented as Cartesian tensors. However, when solving the orientational problem, it is more natural to use the set of spherical functions. Choosing spherical coordinates for the unit vectors e, it, and h as (Q, cp), (0,0), (v(/,0), respectively, that is, taking as the polar axis of the framework, one gets... 

Probability Distribution Functions, Moments and Characteristic Functions... 

The following equations were used to calculate the average distribution function of the degree of product polymerisation in the output of an L-length reactor, with distribution function moments Jq-Jp... 

To solve numerically the linearized kinetic Eq. 24 with the boundary condition (35), a set of values of the velocity c, is chosen. The collision operator Lh is expressed via the values hi x) = h x,Ci). Thus, Eq. 24 is replaced by a system of differential equations for the functions hi x), which can be solved numerically by a finite difference method. First, some values are assumed for the moments being part of the collision operator. Then, the distribution function moments are calculated in accordance with Eqs. 30-34 using some quadrature. The differential equations are solved again with the new moments. The procedure is repeated up to the convergence. 

The average nonuniform permeability is spatially dependent. For a homogeneous but nonuniform medium, the average permeability is the correct mean (first moment) of the permeability distribution function. Permeability for a nonuniform medium is usually skewed. Most data for nonuniform permeability show permeability to be distributed log-normally. The correct average for a homogeneous, nonuniform permeability, assuming it is distributed log-normally, is the geometric mean, defined as ... 

The moments describe the characteristics of a sample or distribution function. The mean, which locates the average value on the measurement axis, is the first moment of values measured about the origin. The mean is denoted by p for the population and X for the sample and is given for a continuous random variable by... 

Frequency analysis is an alternative to moment-ratio analysis in selecting a representative function. Probability paper (see Figure 1-59 for an example) is available for each distribution, and the function is presented as a cumulative probability function. If the data sample has the same distribution function as the function used to scale the paper, the data will plot as a straight line. 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

The hydrodynamic equations are a set of five equations involving the five simple moments of the distribution function, n (or />), v ... 

The moments of the distribution function can be amply related to the expansion coefficients. Using the fact that ( ) andF (0,9>) are unity, we have for the number density ... 

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

The interpretation of the higher-order moments an is simplified if they are first centered about the first moment. To this end, we define the wth central moment pn of the distribution function or, equivalently,... 

The most important characteristic of self information is that it is a discrete random variable that is, it is a real valued function of a symbol in a discrete ensemble. As a result, it has a distribution function, an average, a variance, and in fact moments of all orders. The average value of self information has such a fundamental importance in information theory that it is given a special symbol, H, and the name entropy. Thus... 

Mutual information is thus a random variable since it is a real valued function defined on the points of an ensemble. Consequently, it has an average, variance, distribution function, and moment generating function. It is important to note that mutual information has been defined only on product ensembles, and only as a function of two events, x and y, which are sample points in the two ensembles of which the product ensemble is formed. Mutual information is sometimes defined as a function of any two events in an ensemble, but in this case it is not a random variable. It should also be noted that the mutual... 

The characteristic function and the moment generating function are important tools for computing moments of distributions, studying limits of sequences of distributions, and finding the distribution of sums of independent variables. If Z — X + F, where X and T are independently distributed according to the distribution functions F(x) and 6( y) respectively, the distribution function of Z is given by... 

However, case (ii) above, where there is biaxial symmetry of the distribution function, but no preferred orientation of the structural units about their Ox3 axes is a feasible proposition. Kashiwagi et al.10) and later Cunningham et al. n) have given expressions for the second moment... 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

As different sources are considered, the statistical properties of the emitted field changes. A random variable x is usually characterized by its probability density distribution function, P x). This function allows for the definition of the various statistical moments such as the average. 

Equation (15.39) allows moments of a distribution to be calculated from the Laplace transform of the dilferential distribution function without need for finding f t). It works for any f t). The necessary algebra for the present case is formidable, but finally gives the desired relationship ... 

The same problem can be considered in the opposite direction [34]. Knowing the K-value for a compound (e.g., oxalic acid dihydrate) under specified hydro-dynamic conditions and the fraction undissolved as a function of time, the moments of the distribution function of a dimension of significance can be obtained. 

Distributed Electrostatic Moments Based on the Electron Localization Function Partition... 

Additional software has been developed to merge data from various data collection steps and to model the data using suitable statistical distribution functions. We are working on software to perform corrections for absorption, specimen shape, and misalignment. Library routines for 2-diraensional data smoothing and integration are being adapted to the calculation of orientation functions and other moments of the probability distributions. 

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