Probability value

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Millikan R A A new modification of the cloud method of determining the elementary electrical charge and the most probable value of that charge Phil. Mag. 19 209-28... 

The most probable value of the speed v p can be obtained by differentiation of the distribution function and setting dG(v)/dv = 0 (Kauzmann, 1966 Atkins 1990) to obtain... 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

In addition to and r nis ai other way of characterizing coil dimensions is to consider which end-to-end distance has the greatest probability of occurring for specified n and 1 values. Derive an expression for this most probable value of r, r, from Eq. (1.44). Compare the ratio r ms/ m the ratio from the kinetic molecular theory of gases (consult, say,... 

With a cost of capital of 10 percent the various cash flows can be discounted and summed. Thus for the base cases Z Af = 2,815,600, Z Ajp/d = 754,716, Z Aofd = 614,457, and Z C c/d = 61,446. With corporate taxes payable at 50 percent the aftertax cash flows of the first three items are (1 — 0.50) of the sums calculated above. The discounted working capital and the fixed-capital outlay are not subject to tax. These most probable values are listed and summed in Table 9-11 and, after adjustment for tax, give the modal value of the (NPV) as 276,224. 

Frequency Phase 3 Use Branch Point Estimates to Develop a Ere-quency Estimate for the Accident Scenarios. The analysis team may choose to assign frequency values for initiating events and probability values for the branch points of the event trees without drawing fault tree models. These estimates are based on discussions with operating personnel, review of industrial equipment failure databases, and review of human reliability studies. This allows the team to provide initial estimates of scenario frequency and avoids the effort of the detailed analysis (Frequency Phase 4). In many cases, characterizing a few dominant accident scenarios in a layer of protection analysis will provide adequate frequency information. 

Table 10.1 shows the seleetion of parents for mating from the initial population. If a random number generator is used to generate numbers between 0.0 and 1.0, then the eumulative probability values in Table 10.1 is used as follows ... 

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

At present, the most probable value of the reaction constant for the methoxy-dechlorination of 4-chloroquinolines is obtained from Eq. 

Wahrscheinlichkeits-gesetz, n. probability law. -kurve, /. probability curve, -rechnung, /. calculus of probabilities, -wert, m. probable value. 

The qualitative behavior of this system is most conveniently studied by looking at space-time mai)S of the normalized probability values Pj(t) = diagonal terms is observed by varying S. ... 

It was pointed out above that, if we choose any two acids at random from Table 9, we are not likely to find any correlation between the position of the maximum of K and the value of K itself. But this would not be true, if we compare two acids, paying due regard to the probable value of J non. Three dissociation constants in Table 9 have values smaller than 10 7 and in all three cases the value of 0 falls above 40°. This is doubtless because, in aqueous solution, such a low degree of... 

The accuracy of a determination may be defined as the concordance between it and the true or most probable value. It follows, therefore, that systematic errors cause a constant error (either too high or too low) and thus affect the accuracy of a result. For analytical methods there are two possible ways of determining the accuracy the so-called absolute method and the comparative method. 

The relative error is the absolute error divided by the true value it is usually expressed in terms of percentage or in parts per thousand. The true or absolute value of a quantity cannot be established experimentally, so that the observed result must be compared with the most probable value. With pure substances the quantity will ultimately depend upon the relative atomic mass of the constituent elements. Determinations of the relative atomic mass have been made with the utmost care, and the accuracy obtained usually far exceeds that attained in ordinary quantitative analysis the analyst must accordingly accept their reliability. With natural or industrial products, we must accept provisionally the results obtained by analysts of repute using carefully tested methods. If several analysts determine the same constituent in the same sample by different methods, the most probable value, which is usually the average, can be deduced from their results. In both cases, the establishment of the most probable value involves the application of statistical methods and the concept of precision. 

Test at the 5 per cent probability value if the new method mean is significantly different from the standard reference mean. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Figure 14 shows the displacement of the distribution function towards high / , i.e. the uncoiling of molecules under the influence of stretching for polyethylene (A = 3 x 10-9 m, N = 100 and T = 420 K). This displacement will be characterized by the position of the maximum of the distribution curve, the most probable value of / , i.e. j3m, as a function of x (Fig. 15). Figure 15 also shows the values of stresses a that should be applied to the melt to attain the corresponding values of x (o = xkT/SL, where S is the transverse cross-section of the molecule). 

The most probable values of the parameters are those for which R is at a minimum, or... 

Figure 17. Plot of the potential of zero charge, EaJi, vs. the electron work function, . The point is the most probable value. Data for E0wq from Ref. 140 for Au (111) from Ref. 25 for Pt (111) from Ref. 867.

Two different methods for estimating the probable value of force constant have so far been proposed in estimating the second term in braces in Eq. (11), Binsch et have used the bond-bond polarizabilities, while Nakajima et have made an extensive use of... 

Given that the assumption of normally distributed data (see Section 1.2.1) is valid, several useful and uncomplicated methods are available for finding the most probable value and its confidence interval, and for comparing such results. 

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

Procedure Use the algorithm given below because of the symmetry of the function, only the part 0 < CP < 0.5 is defined cumulative probability values CP lower than 0.5 are transformed to their (decadic) logarithm, the others are first subtracted from 1.00. The sign is appropriately set to 1 or +1. 

Assignments I, J XO H A, P FNZ indices data vector scaling factor for ordinate intermediate results function to transform %-probability values to NPS-scale x is the independent variable in the polynomial. (See Section 5.1.1.)... 

Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

The surface potential of water, x , as any other surface potential, is not measurable. Its probable value is inferred from indirect observations. Such a potential difference has been postulated because the tetrahedral charge... 

According to Eq. (5) the most probable value of x occurs at x = 0. As X increases in magnitude, W(x) decreases monotonically from its maximum at x = 0, the decrease being the more rapid the smaller n. Eq. (5) obviously becomes invalid at values of x approaching nl W x) remains finite, though very small, according to Eq. (5) even for x>nlj where it necessarily is zero. 

The most probable value r p of r for the Is state is found by setting the derivative of D Q(r) equal to zero... 

The ionic potentials can be experimentally determined either with the use of galvanic cells containing interfaces of the type in Scheme 7 or electroanalytically, using for instance, polarography, voltammetry, or chronopotentiometry. The values of and Aj f, obtained with the use of electrochemical methods for the water-1,2-dichloroethane, water-dichloromethane, water-acetophenone, water-methyl-isobutyl ketone, o-nitrotol-uene, and chloroform systems, and recently for 2-heptanone and 2-octanone [43] systems, have been published. These data are listed in many papers [1-10,14,37]. The most probable values for a few ions in water-nitrobenzene and water-1,2-dichloroethane systems are presented in Table 1. 

It is reasonable to assume that the most probable values of the parameters have normal distributions with means equal to the values that were obtained from well test and core data analyses. These are the prior estimates. Each one of these most probable parameter values (kBj, j=l,...,p) also has a corresponding standard deviation parameter estimate. As already discussed in Chapter 8 (Section 8.5) using maximum likelihood arguments the prior information is introduced by augmenting the LS objective function to include... 

Step 3. Provide estimates of the most probable values of the parameters and their standard deviation (k, okj, j=l,...,p). 

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