Gaussian distribution expression

Several investigations have used the Gaussian distribution expressed as follows ... 

Column Efficiency. Under ideal conditions the profile of a solute band resembles that given by a Gaussian distribution curve (Fig. 11.1). The efficiency of a chromatographic system is expressed by the effective plate number defined from the chromatogram of a single band. 

Parameter k of Equation (4.10) is an expression of the breadth of the Gaussian distribution of the cumulative micropore volume IF over the normalized work of adsorption sfifi, and is therefore determined by the pore structure. Thus B also (cf. Equation (4.13)) is characteristic of the pore structure of the adsorbent, and has accordingly been termed the structural constant of the adsorbent. ... 

For a continuous distribution, summation may be replaced by integration and by assuming a Gaussian distribution of size, Stoeckli arrives at a somewhat complicated expression (not given here) which enables the total micropore volume IFo, a structural constant Bq and the spread A of size distribution to be obtained from the isotherm. He suggests that Bq may be related to the radius of gyration of the micropores by the expression... 

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... 

A well-known statistieal distribution is the normal or Gaussian distribution and is expressed by... 

The way forward was proposed by Berne and Pechukas [11] many years later. Their important idea was to consider the overlap between two prolate ellipsoidal gaussian distributions. From the expression for this overlap they evaluated a range parameter which was taken to be the contact distance g and a strength parameter which was set equal to the well depth, e. If the orientations of the two rod-like molecules in the laboratory frame are represented by the unit vectors Ui and Uj and the orientation of the intermolecular vector by the unit vector f then the expression for the angular dependence of the contact distance is... 

In certain applications it is of interest to express the width of the Gaussian distribution at half its value at the maximum. Thus, as the maximum value is /(0) = y/ct jn, the value of z at half of this value is y/ln 2/a and the width at half-height is given by... 

FIGURE 9.1 Linewidth versus standard deviation. A gaussian distribution of unit amplitude is plotted on an x-axis scale in units of the standard deviation (or sigma). At 3.15x0 the unit intensity has dropped to 0.1%. The linewidth in simulations is usually expressed as the halfwidth at half height (HWHH), which is equal to circa 1.17x0, or as twice this value that is, the full width at half height (FWHH). 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

In lack of analytical or numerical methods to obtain the spectra of complicated Hamiltonians, Wigner and Dyson analyzed ensembles of random matrices and were able to derive mathematical expressions. A Gaussian random matrix ensemble consists of square matrices with their matrix elements drawn from a Gaussian distribution... 

Eq. 17.42 is the expression of the resolution for CE in electrophoretic terms. However, the application of this expression for the calculation of Rs in practice is limited because of D,. The diffusion coefficient of different compounds in different media is not always available. Therefore, the resolution is frequently calculated with an expression that employs the width of the peaks obtained in an electropherogram. This way of working results in resolution values that are more realistic as all possible variances are considered (not only longitudinal diffusion in Eq. 17.42). Assuming that the migrating zones have a Gaussian distribution, the resolution can be expressed as follows ... 

The injection of the sample will theoretically result in a square-sided zone which will be broadened by mixing with the solvent to produce a trace which approximates to a Gaussian distribution about the mean. Assuming that the base of the original sample zone is small, the extent of this peak broadening may be expressed as the variance (a2) and the base of the curve (W) will equal 4tr (Figure 3.10). 

Problems arise to get informations about the diffusion coeffients Ky and Kz. If equation (3.4) is interpreted as Gaussian distribution, a lot of available dispersion data can be taken into consideration because they are expressed in terms of standard deviations of the concentration distribution. Though there is no theoretical justification the Gaussian plume formula is converted to the K-theory expression by the transformation /11/... 

As is shown in Eqns. 2-48 and 2-49, the probability density W(e) of electron energy states in the reductant or oxidant particles is represented as a normal distribution function (Gaussian distribution) centered at the most probable electron level (See Fig. 2-39.) as expressed in Eqns. 8-10 and 8-11 ... 

Extinction features that strongly depend on particle size will be obscured, if not totally obliterated, in a polydispersion. Many analytical expressions for the radius probability distribution have been used in Mie calculations. For purposes of illustration we have chosen the Gaussian distribution, according to which the probability that a sphere has radius between a and a + da is... 

On the chromatogram, a represents the width of the peak at 60.6% of its height and /R the retention time. tR and a have to be measured using the same units (time, distance, or eluted volume if the flow is constant). If a is expressed in units of volume (using the flow), then 4Gaussian distribution, equation (1.8) is preferred. Equation (1.9) is less used because of the distortion of most peaks at the base. 

This expression shows that the distribution of emitted 7-ray energies follows a Gaussian distribution with a variance something like... 

The most common reason for a lack of repeatability is the existence of random fluctuations in the environment surrounding the instrument xit possibly in its power supply x2, and also in its input signal x due to random variations in the operation of a device upstream of the instrument in question. These random fluctuations frequently display a normal or Gaussian distribution. The output of a device in response to such random fluctuations may then be expressed as ... 

GRBs emit photons in pulses containing photons with a combination of different wavelengths, whose sources are believed to be ultrarelativistic shocks with Lorentz factor y = (9(100) [28]. Let us consider a wavepacket of photons emitted with a Gaussian distribution in x at the time t = 0. One has to find out how such a pulse would be modified at the observation point at a subsequent time t, because of the propagation through the spacetime foam, as a result of the refractive index. This is similar to the motion of a wavepacket in a conventional dispersive medium. The Gaussian wavepacket may be expressed at t 0 as the real part of... 

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