Gaussian probability function

The distribution of adsorptive potentials of the adsorbent surface is again taken as the Gaussian probability function ... 

Hansch and Fujita used a Gaussian probability function to characterize the partitioning Step 1 and the Hammett function (log(k/k0) = pa) to describe the rate Step 2 in their model (10,12,13). By appropriate mathematical treatment, they arrived at the following general structure-activity relationship which has come to be termed the Hansch Equation. 

Figure 5.22 Scattering intensity I (q) from a stack of parallel lamellae of alternating phases A and B, in which the thicknesses of the lamellae vary according to Gaussian probability functions. Solid line 0a = 0.3, aa = 0.15da, crb = 0.15db. Broken line 0a = 0.3, cra = 0.3da, crb = 0.3d y.

The data are distributed about the expected value according to the normal or gaussian probability function. 

This method was successful in the first attempts to solve equation (15) using a local isotherm equation that allowed adsorbate lateral interactions within patches. The ingenious graphical method, now referred to as the Ross and Olivier method of analysis, has been described in their monograph and in a condensed account by Ross. The technique used by Ross and Olivier employs a Gaussian probability function for the distribution of adsorption energies, i.e. a two-parameter generalized distribution of the form ... 

Zimm s model (1956) is also a chain of beads connected by ideal springs. The chain consists of N identical segments joining + 1 identical beads with complete flexibility at each bead. Each segment, which is similar to a submolecule, is supposed to have a Gaussian probability function. The major difference between the two models lies in the interaction between the individual beads. In the Rouse model, such interaction is ignored in Zimm s model, such interaction is taken into consideration. 

For both independence and finite variance of the involved random variables, the central limit theorem holds a probability distribution gradually converges to the Gaussian shape. If the conditions of independence and finite variance of the random variables are not satisfied, other limit theorems must be considered. The study of limit theorems uses the concept of the basin of attraction of a probability distribution. All the probability density functions define a functional space. The Gaussian probability function is a fixed point attractor of stochastic processes in that functional space. The set of probability density functions that fulfill the requirements of the central limit theorem with independence and finite variance of random variables constitutes the basin of attraction of the Gaussian distribution. The Gaussian attractor is the most important attractor in the functional space, but other attractors also exist. 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

The normal (Gaussian) distribution is the most frequently used probability function and is given by... 

Under specific circumstances, alternative forms for ks j have been proposed like the parabolic or the truncated Gaussian probability distribution function for example [154]. 

Of the several approaches that draw upon this general description, radial basis function networks (RBFNs) (Leonard and Kramer, 1991) are probably the best-known. RBFNs are similar in architecture to back propagation networks (BPNs) in that they consist of an input layer, a single hidden layer, and an output layer. The hidden layer makes use of Gaussian basis functions that result in inputs projected on a hypersphere instead of a hyperplane. RBFNs therefore generate spherical clusters in the input data space, as illustrated in Fig. 12. These clusters are generally referred to as receptive fields. 

It is evident from this that the most likely terminal position is x(x, x) = —Q(i)Sx, as expected from the definition of the correlation function, and the fact that for a Gaussian probability means equal modes. This last point also ensures that the reduction condition is automatically satisfied, and that the maximum value of the second entropy is just the first entropy,... 

A normal (gaussian) probability density function in one centered and standardized variable X reads... 

The Gaussian concept can be extended beyond that already developed in Section IV. The general Gaussian probability density function for the position of a fluid particle released from a source located at (x y, z ) at time t can be expressed as (Lamb, 1980)... 

While radioactive decay is itself a random process, the Gaussian distribution function fails to account for probability relationships describing rates of radioactive decay Instead, appropriate statistical analysis of scintillation counting data relies on the use of the Poisson probability distribution function ... 

For the case of D-RADP-20 we have assumed a Gaussian probability distribution of transition temperatures, and therefore used an error function to fit the transition region. For D-RADP-25 the transition temperatures range from 138 K down to 118 K. In this temperature range, it is not possible to separate... 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

If the data to be fit are continuous there are general nonlinear methods which can be used to fit almost any probability function (8), including a variety of so-called probit analyses for (assumed) Gaussian data (9). For many of these methods, convergence is slow or nonexistant if the values initially selected for the fitted parameters are not sufficiently close to the final values. 

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