BE Statistics

When T 0, Hj 0 in BE statistics if Cj i. If (Cj - (i)/kT 0 from above, which happens at a low temperature, Uj - oo. This means that all states end up in the lowest state when the temperature is decreased to zero. 

In FD statistics, Oj varies between 1 and 0 as (Cj - p)/kT varies between -oo and +00. The electrons in a metal will occupy only the levels that are below the Fermi level at T 0. If e p and (e - p) kT and increasing, the different distributions will begin to agree. 

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Equation (2.28), being statistical in nature, requires a large number of particles to be measured, especially if the spread of particle size is wide. The possibility of error from this source is stressed by Arnell and Henneberry who found that in a particular sample of finely ground quartz, two particles in a total of 335 had a diameter about twenty times the most probable diameter, and that if these were overlooked the calculated value of A would be nearly doubled. 

Various methods have been developed to eliminate biases which otherwise can skew results. The wines must be presented without identification, although the taster should be told the type of wines (the best strawberry wine should rate very poorly in a Cabernet class). Eor the most informative results, many details of coding, presentation order, repHcation, etc must be considered. The results must be statistically examined to estimate whether or not they could have been obtained accidentally. Statistical analysis is an entire field in and of itself, and wine studies have contributed greatly to its present sophistication, as appHed in the flavor field. 

Tests on Cable Constructions. The Association of Edison Illumination Companies (AEIC) has approved an accelerated cable hfe test in which typical underground distribution power cables can be statistically compared based on their resistance to water treeing (number of days to fail). The comparison can be made by varying the type of insulation and/or other cable layers in an environment that contains hot water (90°C) under 8V/fi (200 V/mil) voltage stresses (four times the typical power cables operating voltages). 

Nature When individuals or items are obseiwed and classified accordingto two different criteria, the resultant counts can be statistically ana ed. For example, a market suiwey may examine whether a new product is preferred and if it is preferred due to a particular characteristic. 

Count data, based on a random selection of individuals or items which are classified according to two different criteria, can be statistically analyzed through the distribution. The purpose of this analysis is to determine whether the respective criteria are dependent. That is, is the product preferred because of a particular characteristic ... 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

When the silica surface is in contact with a solvent, the surface is covered with a layer of the solvent molecules. If the mobile phase consists of a mixture of solvents, the solvents compete for the surface and it is partly covered by one solvent and partly by the other. Thus, any solute interacting with the stationary phase may well be presented with two, quite different types of surface with which to interact. The probability that a solute molecule will interact with one particular type of surface will be statistically controlled by the proportion of the total surface area that is covered by that particular solvent. 

Warshawsky and coworkers have recently reported the synthesis of a class of compounds which they call polymeric pseudocrown ethers . A chloromethylated polystyrene matrix is used here as in 6.6.2, but instead of adding a crown to the backbone, a strand of ethyleneoxy units is allowed to react at two different positions on the chain, thus forming a crown. Such systems must necessarily be statistical, and the possibility exists for forming interchain bridges as well as intrachain species. Nevertheless, polymers which could be successfully characterized in a variety of ways were formed. A schematic representation of such structures is illustrated below as compound 30. ... 

You should review the contract and the detail specifications to identify whether your existing controls will regulate quality within the limits required. You may need to change the limits, the standards, the techniques, the methods, the environment, and the instruments used to measure quality characteristics. One technique may be to introduce Just-in-time as a means of overcoming storage problems and eliminating receipt inspection. Another technique may be Statistical Process Control as a means of increasing the process yield. The introduction of these techniques needs to be planned and carefully implemented. 

Like all other methods for analyzing censored failure data, the hazard plotting method is also based on a certain assumption that must be satisfied if we are going to rely on the results. The assumption is that if the unfailed units were mn to failure, their failure times would be statistically independent of their censoring times. In other words, there is no relationship or correlation between the censoring time of a unit and the failure time. For example. 

When this relationship holds, X(t + r) and Y(t + r ) are said to be statistically independent. Statistical independence will be discussed in detail in Section 3.10. 

The converse of this statement is not true in general random variables satisfying (3-179) need not be statistically independent. 

Finally, an infinite set of random vectors is defined to be statistically independent if all finite subfamilies are statistically independent. Given an infinite family of identically distributed, statistically independent random vectors having finite means and covariances, we define their normalized sum to be the vector sfn, , sj where... 

We conclude this section by deriving an important property of jointly gaussian random variables namely, the fact that a necessary and sufficient condition for a group of jointly gaussian random variables 9i>- >< to be statistically independent is that E[cpjCpk] = j k. Stated in other words, linearly independent (uncorrelated),46 gaussian random variables are statistically independent. This statement is not necessarily true for non-gaussian random variables. 

Now consider N pairs of random variables each having the distribution above, and each pair being statistically independent of all other pairs. Define w = 2 -t wn> = 2 -i and define HN(r,t) to be the joint moment generating function of w,z. 

When collisional effects are negligible the second component in (3.8) vanishes, and J becomes a constant. The solution obtained in this case is used in Eq. (3.4). If collisions occur they change J and frustrate the vibrational phase simultaneously. Nevertheless, the processes are usually considered to be statistically independent ... 

Room-temperature fluorescence (RTF) has been used to determine the emission characteristics of a wide variety of materials relative to the wavelengths of selected Fraunhofer lines in support of the Fraunhofer luminescence detector remote-sensing instrument. RTF techniques are now used in the compilation of excitation-emission-matrix (EEM) fluorescence "signatures" of materials. The spectral data are collected with a Perkin-Elraer MPF-44B Fluorescence Spectrometer interfaced to an Apple 11+ personal computer. EEM fluorescence data can be displayed as 3-D perspective plots, contour plots, or "color-contour" images. The integrated intensity for selected Fraunhofer lines can also be directly extracted from the EEM data rather than being collected with a separate procedure. Fluorescence, chemical, and mineralogical data will be statistically analyzed to determine the probable physical and/or chemical causes of the fluorescence. 

Alternatively, the experimental error can be given a particular value for each reaction of the series, or for each temperature, based on statistical evaluation of the respective kinetic experiment. The rate constants are then taken with different weights in further calculations (205,206). Although this procedure seems to be more exact and more profoundly based, it cannot be quite generally recommended. It should first be statistically proven by the F test (204) that the standard errors in fact differ because of the small number of measurements, it can seldom be done on a significant level. In addition, all reactions of the series are a priori of the same importance, and it is always a... 

Figure 1.20. Monte Carlo simulation of 25 normally distributed measurements raw data are depicted in panel A, the derived means Xmean CL(Xmean) in B, and the standard deviation % + CL( t) in C. Notice that the mean and/or the standard deviation can be statistically different from the expected values, for instance in the range 23 < n < 25 in this example. The ordinates are scaled in units of la. 

The first theoretical attempts in the field of time-resolved X-ray diffraction were entirely empirical. More precise theoretical work appeared only in the late 1990s and is due to Wilson et al. [13-16]. However, this theoretical work still remained preliminary. A really satisfactory approach must be statistical. In fact, macroscopic transport coefficients like diffusion constant or chemical rate constant break down at ultrashort time scales. Even the notion of a molecule becomes ambiguous at which interatomic distance can the atoms A and B of a molecule A-B be considered to be free Another element of consideration is that the electric field of the laser pump is strong, and that its interaction with matter is nonlinear. What is needed is thus a statistical theory reminiscent of those from time-resolved optical spectroscopy. A theory of this sort was elaborated by Bratos and co-workers and was published over the last few years [17-19]. 

Several terms have been used to define LOD and LOQ. Before we proceed to develop a uniform definition, it would be useful to define each of these terms. The most commonly used terms are limit of detection (LOD) and limit of quantification (LOQ). The 1975 International Union of Pure and Applied Chemistry (lUPAC) definition for LQD can be stated as, A number expressed in units of concentration (or amount) that describes the lowest concentration level (or amount) of the element that an analyst can determine to be statistically different from an analytical blank 1 This term, although appearing to be straightforward, is overly simplified. If leaves several questions unanswered, such as, what does the term statistically different mean, and what factors has the analyst considered in defining the blank Leaving these to the analyst s discretion may result in values varying between analysts to such an extent that the numbers would be meaningless for comparison purposes. 

Because the sampling plan is central to the overall objective of an LSMBS, the plan should be statistically designed. The overall intent of the design is that... 

Net recoveries of cyfluthrin from matrices fortified at 0.01-5.05 mg kg ranged from 77 to 119%. The limit of detection (LOD) is defined as the lowest concentration that can be determined to be statistically different from a blank or control. Calculate the value by taking the standard deviation of the residue values from the analysis of the recovery samples at the limit of quantification (LOQ) and using the equation... 

The requirement for such spatially averaged measurements is that the granular assembly be statistically stable, that is, the probability of finding grains in a particular distribution of situations, such as having certain velocities or of it being in a certain set of neighbor distances, is independent of time. 

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