Conditional variance

Therefore, the maximum likelihood estimator of X is (1 + x )/ y and the asymptotic variance, conditional on... 

A method has been described which utilizes minimum-variance conditions and which leads to almost complete uncoupling of kinetic variables which further reduces the uncertainty in the estimates of the kinetic parameters. 

Precision is a measure of the spread of data about a central value and may be expressed as the range, the standard deviation, or the variance. Precision is commonly divided into two categories repeatability and reproducibility. Repeatability is the precision obtained when all measurements are made by the same analyst during a single period of laboratory work, using the same solutions and equipment. Reproducibility, on the other hand, is the precision obtained under any other set of conditions, including that between analysts, or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the reproducibility of an analysis can be no better than its repeatability. 

The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 

Hydrostatic drives allow for selection of any travel speed up to the maximum without a concurrent variance in engine speed. The engine can be operated at the governed speed to provide proper operating speeds for auxiliary elements, eg, the threshing section of a combine. A frill range of travel speeds is available to adjust to terrain or crop conditions. Industrial applications for hydraulic systems and hydrostatic transmissions include the following (16) ... 

Semiconducting Ceramics. Most oxide semiconductors are either doped to create extrinsic defects or annealed under conditions in which they become non stoichiometric. Although the resulting defects have been carefully studied in many oxides, the precise nature of the conduction is not well understood. Mobihty values associated with the various charge transport mechanisms are often low and difficult to measure. In consequence, reported conductivities are often at variance because the effects of variable impurities and past thermal history may overwhelm the dopant effects. 

The direct-labor-cost variance can, if necessary, be broken down into a direc t-labor-idle-time variance in addition to the direct-wage-rate and direct-labor-efficiency variances. The direc t-labor-idle-time variance is simply the number of idle labor-hours in the period multiplied by the standard wage rate. This is rarely relevant to the conditions existing in process plants except when maintenance is involved. 

The Erlang (or gamma) and dispersion models can be related by equating the variances of their respective E(E) functions. The result for the closed-ends condition is... 

Are there any zoning, variance or conditional use regulations associated with the site that may restrict or delay production of the material If yes, explain. 

On page 4, rates are calculated for the four specified conditions. Variance is calculated in the experimental results and correlation coefficients are used to show that fraction of the variance in the experimental results accounted for by the model. This is over 99%. Finally the experimental error is calculated from the repeated experiments on page 5. 

Using the first and second order terms in the variance equation gives exactly the same answer. For different conditions, say where one variable is not dominating the situation as above for the load, then the use of the variance equation with second order terms will be more effective. 

All of the tests were with code conditions except for CO2 where the Mach number variance was more than plus 5%. The results are shown in Figure 10-4. It was concluded that the R12 test provided the more nearb/... 

To confirm the pertinence of a particular dispersion equation, it is necessary to use extremely precise and accurate data. Such data can only be obtained from carefully designed apparatus that provides minimum extra-column dispersion. In addition, it is necessary to employ columns that have intrinsically large peak volumes so that any residual extra-column dispersion that will contribute to the overall variance is not significant. Such conditions were employed by Katz et al. (E. D. Katz, K. L. Ogan and R. P. W. Scott, J. Chromatogr., 270(1983)51) to determine a large quantity of column dispersion data that overall had an accuracy of better than 3%. The data they obtained are as follows and can be used confidently to evaluate other dispersion equations should they appear in the literature. 

Variances in resin performance and capacities can be expected from normal annual attrition rates of ion-exchange resins. Typical attrition losses that can be expected include (1) Strong cation resin 3 percent per year for three years or 1,000,000 gals/ cu.ft (2) Strong anion resin 25 percent per year for two years or 1,000,000 gals/ cu.ft (3) Weak cation/anion 10 percent per year for two years or 750,000 gals/ cu. ft. A steady falloff of resin-exchange capacity is a matter of concern to the operator and is due to several conditions ... 

Two types of boundary conditions are considered, the closed vessel and the open vessel. The closed vessel (Figure 8-36) is one in which the inlet and outlet streams are completely mixed and dispersion occurs between the terminals. Piston flow prevails in both inlet and outlet piping. For this type of system, the analytic expression for the E-curve is not available. However, van der Laan [22] determined its mean and variance as... 

Table 8-8 summarizes the results of the cases discussed above including the boundary conditions, the expression for C(6) at z = 1, and the mean and variance for C(6). 

Interim order An official statement issued by OSHA allowing an employer to continue operations under existing conditions while an application for a variance is being considered. 

If we improve process monitoring and data logging, we ll be better equipped to track and predict variances in process operating conditions. 

The ability of a GC column to theoretically separate a multitude of components is normally defined by the capacity of the column. Component boiling point will be an initial property that determines relative component retention. Superimposed on this primary consideration is then the phase selectivity, which allows solutes of similar boiling point or volatility to be differentiated. In GC X GC, capacity is now defined in terms of the separation space available (11). As shown below, this space is an area determined by (a) the time of the modulation period (defined further below), which corresponds to an elution property on the second column, and (b) the elution time on the first column. In the normal experiment, the fast elution on the second column is conducted almost instantaneously, so will be essentially carried out under isothermal conditions, although the oven is temperature programmed. Thus, compounds will have an approximately constant peak width in the first dimension, but their widths in the second dimension will depend on how long they take to elute on the second column (isothermal conditions mean that later-eluting peaks on 2D are broader). In addition, peaks will have a variance (distribution) in each dimension depending on... 

This is an equation which fixes the relation existing between the number of phases (/ ), the number of components ( i), and the variance, or number of degrees of f reedom (F), of a heterogeneous system in equilibrium, subject to certain conditions which are usually satisfied in practice. The rule states that... 

Though the kinetic results above can be rationalised by reasonable premises, one experimental observation is markedly at variance and this is that the percentage of orr/io-benzoylation of toluene is constant under all conditions clearly there is still much to be understood about the role of the catalyst in these reactions. 

To determine the band dispersion that results from a significant, but moderate, sample volume overload the summation of variances can be used. However, when the sample volume becomes excessive, the band dispersion that results becomes equivalent to the sample volume itself. In figure 10, two solutes are depicted that are eluted from a column under conditions of no overload. If the dispersion from the excessive sample volume just allows the peaks to touch at the base, the peak separation in milliliters of mobile phase passed through the column will be equivalent to the sample volume (Vi) plus half the base width of both peaks. It is assumed in figure 10 that the efficiency of each peak is the same and in most cases this will be true. If there is some significant difference, an average value of the efficiencies of the two peaks can be taken. 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

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