Moments of distribution function

For a continuous distribution function f(t), the kth moment about the origin is defined by  

The zeroth moment (k = 0) is simply the area under the distribution curve  

The first moment (k = 1) is the mean of the distribution, t, a measure of the location of the distribution, or the expected (average) value (t) of the distribution f(t)  

Higher moments are usually taken about the mean to characterize the shape of the distribution about this centroid. 

The second moment (k = 2) about the mean is called the variance, of, and is a measure of the spread of the distribution  

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Using the relation between second moments of distribution functions f R,o) and... 

Generating functions are used in calculating moments of distributions for power series expansions. In general, the nth moment of a distribution,/fxj is E x ") = lx" f x) dx, where the integration is over the domain of x. (If the distribution is discrete, integration is replaced by summation.)... 

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... 

Some fundamental definitions and properties of distribution functions are summarized briefly in this section. The most important statistical weights, averages, and moments frequently encountered in polymer analysis are introduced [7], Most quantities defined here will feature later again in the discussion of the individual analytical techniques. 

For any quantity that is a function of time we can describe its properties in terms of its distribution function and the moments of this function. We first define the probability distribution function p(t) as the probability that a molecule entering the reactor wih reside there for a time t. This function must be normalized... 

Next, we define a parallel set of NPD function in continuous flow recirculating systems. We restrict our discussion to steady flow systems. Here, as in the case of RTD, we distinguish between external and internal NPD functions. We define fk and 4 as the fraction of exiting volumetric flow rate and the fraction of material volume, respectively, that have experienced exactly k passages in the specified region of the system. The respective cumulative distribution functions, and /, the means of the distributions, the variances, and the moments of distributions, parallel the definitions given for the batch system. 

A number of distribution functions have been identified experimentally for a variety of systems ( 3) and, in particular, the Log-Normal distribution is extensively used for the calculation of the integral and for the evaluation of the moments of the particle size distribution (8—12). The problem with this approach is that, in general, the shape of the particle size distribution is unknown and thus, the average particle diameters obtained are conditional upon... 

Appendices follow Chapter 6. In Chapter 2, it has been pointed that local entropy may be expressed in terms of same independent variables as if the system were at equilibrium (local equilibrium). The limitations of Gibbs equation have been discussed in Appendix I. At no moment, molecular distribution function of velocities or of relative positions may deviate strongly from their equilibrium form. This is a sufficient condition for the application of thermodynamics method. Some new developments related to alternative theoretical formalism such as extended irreversible are discussed in Appendices II and III. 

Llorente, M. A. Rubio, A. M. Freire, J. J., Moments and Distribution Functions of the End-to-End Distance of Short Poly(dimethylsiloxane) and Poly(oxyethylene) Chains. Application to the Study of Elasticity in Model Networks. Macromolecules 1984,17, 2307-2315. 

Experimental transverse relaxation decays cannot be usually described from exponential time functions but integral treatments of these cmves yield standard parameters equivalent to relaxation rates (81). Without entering into too many details, it may be worth noting that the integral treatment amount to calculating several moments of the function P R). The first and the third moments of the distribution fimction are obtained from the two following integral treatments of the experimental relaxation curves... 

Figure 8 shows examples of distribution functions fijn) = mnJ(mn 3mioL obtained from analytical solutions presented in Table 2 for constant (kij = 2), sum kij — i + j), and product = 2ij) kernels at different moments in time. It is interesting that at Imig times the distribution function for the constant kernel has a bell-like shape, whereas for sum and product kernels they monotonically decay with increasing m. 

For solutions the polarization interaction is mainly responsible for the shift, while the width is mostly given by the electrostatic interaction. This interaction can often be localized to the first solvation shell, for instance, in the case of water and probably for other polar liquids as well. The angular distribution of the molecules in the first solvation shell thus represents the first moment of the energy distribution well, while the longer range electrostatic contribution largely cancels due to disorder. One can see the two terms above as an extension of the potential model with relaxation, where the terms are obtained from integration of distribution functions, which in turn are dependent on temperature and pressure. 

How can this formal treatment of the distribution function (and resulting order parameters) be generalized to include the smectic-A structure We find the clue in Kirkwood s treatment of the melting of crystalline solids. In a crystal the density distribution function (the translational molecular distribution function) is periodic in three dimensions and can be expanded in a three-dimensional Fourier series. Kirkwood does this and then identifies the order parameters of the crystalline phase as the coefficients in the Fourier series. For simplicity let us consider a one-dimensionally periodic structure (such as the smectic-A but with the orientational order suppressed for the moment). The distribution function, which describes the tendency of the centers of mass of molecules to lie in layers perpendicular to the z-direction, can be expanded in a Fourier series ... 

Often, moments of distribution/density functions are used in particulate systems. The th moment of the particle size density function is defined by... 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Moments of a distribution often provide information that can be used to characterize particulate matter. Theyth moment of the population density function n is defined as... 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

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 ... 

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