Distribution functions central moments

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 Characteristic Function.—The calculation of moments is often quite tedious because of difficulties that may be encountered in evaluating the pertinent integrals or sums. This problem can be simplified quite often by calculation of the so-called characteristic function of the distribution from which, as we shall see, all moments can be derived by means of differentiation. This relationship between the characteristic function and moments is sufficient reason for studying it at this time however, the real significance of the characteristic function will not become apparent until we discuss the central limit theorem in a later section. 

Mintzer, David, 1 Mitropolsky, Y. A361,362 Mixed groups, 727 Modality of distribution, 123 Models in operations research, 251 Modification, method of, 67 Molecular chaos, assumption of, 17 Miller wave operator, 600 Moment generating function, 269 Moment, 119 nth central, 120... 

The second moment m2j of a peak i with the distribution function y,(f) is related to the retention time mu and as the central moment represents the peak time variance [Pg.330]

The best way to correct determination of chemisorption activation parameters on heterogeneous oxides surface includes the calculation of its both distribution functions on the activation energy and on logarithms of the pre-exponential factor and their central moments from the kinetic isotherms measured at different temperatures. Obtained from such calculations the isokinetic temperature and logarithm of rate constant at this temperature reflect the gaseous organic compound reactivity toward active sites of the heterogeneous oxides surface. 

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

For the moment let us follow closely the analysis of Richmond [68,69] who considered two planar surfaces, one at z = 0 and one at z = h, with arbitrary nonuniform distributions of charge or potential y s) and y2(s). The boundaries mark the ends of two infinite planar half spaces with dielectric permittivities, g] and S3, separated by a third, intermediate dielectric continuum of width h and permittivity, e2. This central medium contains a simple electrolyte solution. The generic function, y(s), we use to represent either a surface potential, P(s), or a surface charge, cr(s). s = (x,.y) is again the position vector in the plane of a surface. An arbitrary source distribution can be represented by the Fourier integral,... 

An important issue is to verify that the energy differences are normally distributed. Recall that if the moments of the energy difference are bounded, the central limit theorem implies that given enough samples, the distribution of the mean value will be Gaussian. Careful attention to the trial function to ensure that the local energies are well behaved may be needed. 

The above moments of a charge distribution can be related to the statistical moments of random variables. The orbital centroid vector is namely identical to the central first moment M(Q of a random variable , with density function p(x) ... 

Disorder was introduced into this system by postulating a distribution of waiting times. A complementary extension of the theory may be made by considering a distribution of jump distances. It may be shown that, as a consequence of the central limit theorem, provided the single-step probability density function has a finite second moment, Gaussian diffusion is guaranteed. If this condition is not satisfied, however, then Eq. (105) must be replaced by... 

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