Symmetric Gaussian distribution

In the absence of an external force, the probability of moving to a new position is a spherically symmetrical Gaussian distribution (where we have assumed that the diffusion is spatially isotropic). 

Fig. 9. The efTect of the dipole-dipole interaction on inhomogeneously broadened infrared absorption peaks. Without and with the interaction turned on for peaks with I, a low frequency tail, II, a symmetric Gaussian distribution and III, a high frequency tail. (Adaptedfrom Persson...

Another consequence of the harmonic oscillations in the solvent atmosphere around the reacting ion is that the thermal distribution of electronic states in the redox solution follows a symmetrical Gaussian distribution [71]... 

The interpretation of adsorption isotherms in terms of a surface displaying a symmetrical (Gaussian) distribution of adsorptive potentials to the adsorbate has been successfully applied to a number of systems but marked exceptions have been found, particularly with crystals that were selected as likely to have completely homotattic surfaces. These exceptional adsorption isotherms are readily interpreted, however, as the result of a sum of two, or occasionally three, distinctly different Gaussian distributions, presumably deriving from the same number of surface constituents. Among these constituents the expected homotattic substrate, which is associated with a particular crystal face, can always be identified. 

At high values of. v (x 1) the asymmetric Poisson distribution becomes identical with the symmetric Gaussian distribution... 

Without going into too much detail, one can state that, in practice, when Dapp is not too large" (number of stages > 100, Guiochon et al., 1994b) the solution for small injections can be approximated by a symmetrical Gaussian distribution ... 

For the exponentially modified Gaussian, the variances of the symmetrical Gaussian distribution a and the tailing exponential decay are additive ... 

Distribution Coefficients. Gel-permeation stationary-phase chromatography normally exhibits symmetrical (Gaussian) peaks because the partitioning of the solute between mobile and stationary phases is linear. Criteria more sophisticated than those represented in Figure 8 are seldom used (34). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

All the remaining central moments /x2n+1 are equal to zero because the gaussian distribution is symmetric about its mean. 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

It is generally assumed that the disorder can be represented by a symmetric Gaussian-type pair distribution function and that the thermal vibration will be harmonic in nature. 

One can further conclude that that these two Gaussian distributions are symmetrically located on the upper and lower sides of AA, and the free energy difference A A, the mean work W OF, for the forward and — W >0 for the reverse transformation) and the variance of work obey the following relationships ... 

Indeterminate errors arise from the unpredictable minor inaccuracies of the individual manipulations in a procedure. A degree of uncertainty is introduced into the result which can be assessed only by statistical tests. The deviations of a number of measurements from the mean of the measurements should show a symmetrical or Gaussian distribution about that mean. Figure 2.2 represents this graphically and is known as a normal error curve. The general equation for such a curve is... 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

In a situation whereby a large number of replicate readings, not less than 5 0, are observed of a titrimetric equivalence point (continuous variable), the results thus generated shall normally be distributed around the mean in a more or less symmetrical fashion. Thus, the mathematical model which not only fits into but also satisfies such a distribution of random errors is termed as the Normal or Gaussian distribution curve. It is a bell-shaped curve which is noted to be symmetrical about the mean as depicted in Figure 3.2. 

A set of replicate measurements is said to show a normal or Gaussian distribution if it shows a symmetrical distribution about the mean value. 

Data from phase-modulation fluorometry have been analyzed using an alternative approach to those described above, as expounded by Gratton and co-workers(14 12 13,22) and Lakowicz et al. W> Here, Lorentzian or Gaussian distribution functions with widths and centers determined by least-squares analysis are used to model the unknown distribution function. While this approach may introduce assumptions about the shape of the ultimate distribution function since these trial functions are symmetric, it has the advantage of minimizing the number of parameters involved in the fit. Here, a minimum x2 is sought, where... 

Thus the center of each axis equals zero and the distribution around the center becomes symmetrical for Gaussian distributed feature values a (x ) represents the unit length along the axes. 

If subsequent analyses of the bulk sample deviate by more than a predetermined amount, the whole batch of results is rejected. Results are thus only accepted if they fall between specified values of s above and below the mean, where Is includes 68%, 2s includes 95% (the normally accepted value), and 3s includes 99.7% of results. The scatter of results usually assumes a symmetrical normal or Gaussian distribution about the mean, as shown in Figs 12.1 and 12.2. 

This type of behavior would produce distributions of activity shown in Figure 1 where the various peaks are symmetric gaussians of increasing width. 

Figure 1.5—Theoretical plate model. Computer simulation of the elution of two compounds, A and B, chromatographed on a column with 30 theoretical plates (KA = 0.5 KB = 1.6 MA — 300 pg Mb = 300 pg) showing the composition of the mixture at the outlet of the column after the first 100 equilibria. As is evident from the graph, this model leads to a non-symmetrical peak. However, when the number of equilibria is very large, and because of diffusion, the peak looks more and more like a Gaussian distribution.

If an experiment is repeated a great many times and if the errors are purely random, then the results tend to cluster symmetrically about the average value (Figure 4-1). The more times the experiment is repeated, the more closely the results approach an ideal smooth curve called the Gaussian distribution. In general, we cannot make so many measurements in a lab experiment. We are more likely to repeat an experiment 3 to 5 times than 2 000 times. However, from the small set of results, we can estimate the statistical parameters that describe the large set. We can then make estimates of statistical behavior from the small number of measurements. 

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