Theoretical Distribution Functions

The histogram is a graphical device which is both attainable in practice and also an approximation to a theoretical distribution function. 

Thus, one can be far from the ideal world often assumed by statisticians tidy models, theoretical distribution functions, and independent, essentially uncorrupted measured values with just a bit of measurement noise superimposed. Furthermore, because of the costs associated with obtaining and analyzing samples, small sample numbers are the rule. On the other hand, linear ranges upwards of 1 100 and relative standard deviations of usually 2% and less compensate for the lack of data points. 

Various theoretical distribution functions have been proposed, such as normal or Gaussian distribution and the log-normal distribution. The simplest case is... 

Fractionation Data and Distribution Analysis of HEC After One Day of Cellulase Attack. In Table III, the fractionation data of HEC are given after one day of enzymic hydrolysis. As after one hour of enzymic hydrolysis, no theoretical distribution function accorded well with the fractionation data, but we evaluated the parameters by numerical analysis, using the Gauss-Laguerre method (43,44). This method has one advantage over other numerical methods, e.g., (45)—all the calculations involved can be done manually without the need of high-speed computers. 

The distribution of species sensitivities follows a theoretical distribution function. 

For a number of theoretical distribution functions of sphere radii f(r), the authors calculated the distribution of the radii of circles on the sectioned surface, making use of Eq. (53). The result is shown in Fig. 25, (for comparison, the maximum cell diameter, b, was taken as equal in all calculations). 

Note that /to = 1 for this and most other theoretical distribution functions. The mean of the Elory distribution is (obviously) In, and... 

Figure 2.1 Histogram for the measurements of Table 2.1 and the theoretical distribution function according to a Gaussian distribution solid line).

To achieve a theoretical distribution function for the mileage distribution the empirical data has to be fit by means of appropriate parameter estimations. In most cases the lognormal distribution L(s) = LN(iJ,a ) with... 

The application of the Weibull distribution for describing lifetime has been proved in several preceded analysis and was therefore applied to fit the empirical warranty data. The usage of other theoretical distribution functions, such as the lognormal distribution, is also suitable to describe the lifetime but was not considered in this analysis. See (Meyna Pauh 2003) for more detailed information. 

The hazard rate h(t) is another important reliability characteristic which can be used to describe the field failure behaviour when using a Weibull distribution. All three elements of the well known bath-tub-curve can be displayed by this theoretical distribution function. Thus it is possible to describe early failure, random failure and wearout failure. 

As described before nine existing identical machines have been analysed. The results consist of the safety-related empirical and theoretical failure probability of the safety-related function. Further, the failure density function of the theoretical distribution function is calculated and illustrated to clarify how the maximum of the failure density function is obtained. The received maximum values can finally be compared with the quantitative requirements of the EN ISO 13849-1 and the Performance Level can be determined. 

Fig. 21. Differential distribution functions of ultrasonically degraded polystyrene. dm/dP differential distribution function of d ree of polymerization dm = mass of polymer having a degree of polymerizatkm betwee and P+dP initial molecular we My, = 411000, My d = 1.05 solvent tetrahydrofuran ultrasonic power of 48 watts at 20 kc/sec. Curves fe/f are expoi-moitah canesr ht are theoretical distribution functions calculated from pie de ation model. Param H of curves is time of irradiation (in min) iReC. (97)1...

An alternative way to derive the theoretical distribution function Is/ipi is to present the initial stationary nucleation rate la in the form (cf. equations (2.82)-(2.84), Chapter 2.3.2.1) ... 

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