Value, extreme

It is useful to truncate the lines at the extreme values which are considered likely to occur, e.g. oil price may be considered to vary between -40% and +20% of the base case consumption. This presentation adds further value to the plot. 

Fig. 10.19 The probability density of the extreme value distribution typical of the MSP scores for random sequena The probability that a random variable with this distribution has a score of at least x is given by 1 - exp[-e -where u is the characteristic value and A is the decay constant. The figure shows the probability density function (which corresponds to the function s first derivative) for u = 0 and A = 1.

As shown by Examples 4.1 and 4.2, the mean and median provide similar estimates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensitive to measurements with extreme values. For example, introducing the transcription error discussed earlier for the mean only changes the median s value from 3.107eto3.112e. 

Many distribution functions can be apphed to strength data of ceramics but the function that has been most widely apphed is the WeibuU function, which is based on the concept of failure at the weakest link in a body under simple tension. A normal distribution is inappropriate for ceramic strengths because extreme values of the flaw distribution, not the central tendency of the flaw distribution, determine the strength. One implication of WeibuU statistics is that large bodies are weaker than small bodies because the number of flaws a body contains is proportional to its volume. 

In Fig. 3-25 the locational dependence of t/, and is shown together. For practical applications and because of possible disturbance by foreign fields (e.g., stray currents) and t/g are less amenable to evaluation than f/g, which can always be determined by a point of inflection between two extreme values [50]. Furthermore, it should be indicated by Fig. 2-7 that there is a possibility of raising the sensitivity by anodic polarization which naturally is only applicable with small objects. In such cases care must be particularly taken that the counter electrode is sufficiently far away so that its voltage cone does not influence the reference electrodes. 

The last letter indicates the extreme value at which the alarm/switch function is trigged ... 

Gn L) is often difficult to determine for a given load distribution, but when is large, an approximation is given by the Maximum Extreme Value Type I distribution of the maximum extremes with a scale parameter, 0, and location parameter, v. When the initial loading stress distribution,/(L), is modelled by a Normal, Lognormal, 2-par-ameter Weibull or 3-parameter Weibull distribution, the extremal model parameters can be determined by the equations in Table 4.11. These equations include terms for the number of load applications, n. The extremal model for the loading stress can then be used in the SSI analysis to determine the reliability. 

For example, to determine the reliability, R , for independent load applications, we can use equation 4.33 when the loading stress is modelled using the Maximum Extreme Value Type I distribution, as for the above approach. The CDF for the... 

Table 4.11 Extremal value parameters from initial loading stress distributions...

Since elemental RSFs are reasonably similar for electron-gas SNMSd, a standardless analysis will result in compositions accurate to within a factor of 5 for matrices with major element RSFs close to the average, and to within a factor of 25 for matrices with major element RSFs at the extreme values. More importantly. 

There are other distributions that can be used in a variety of reliability models. The Poisson, the extreme value, gamma, binomial, and Rayleigh distributions are sometimes used in specialized models. 

The authors have compared in every case the p-values for the heterocyclic series (ph) with the corresponding p-value for the benzene series (Pb) under identical conditions. The ratios ph/pb f e thiophene series vary between 0.63 and 1.34, with the two extreme values applying to the 5-R-3-Y systems. For the other systems the variation is between 0.83 and 1.20. with an average value of 0.99. The three furan-system values, all for 5-R-2-Y, are more divergent, between 0.89 and 1.39, with an average of 1.15, apparently slightly higher, than in the thiophene system. 

Plotting data on hazard paper requires less effort, and at the same time, it succeeds in using all of the failures and gives the same information. Hazard papers are shown here for exponential, Weibull, normal, log normal, and extreme value distributions. 

Given next are the different methods for estimating distribution parameters on exponential, Weibull, normal, log normal, and extreme-value hazard papers. The methods are explained with the aid of simulated data from known distributions. Thus, we can judge from the hazard plots how well the hazard-plotting method does. 

Finley, H. F., An Extreme-value Statistical Analysis of Maximum Pit Depths and Time to First Perforation , Corrosion, 23, 83 (1%7)... 

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