Logarithms standard deviation

It is assumed that both and u are lognormally distributed with logarithmic standard deviations of and P, j, respectively. The advantages of this formulation are ... 

The size distributions of the fractions were plotted on log-probability paper as particle diameter (in microns) against cumulative percent of particles smaller than the indicated size. Figure 1 shows such a plot for the Johnie Boy size fractions. Such plots were compared for several samples with similar plots on linear-probability paper. Almost always the data could be described better by a lognormal rather than by a normal distribution law, after proper allowance for the presence of a maximum and a minimum size in each fraction. The parameters of the distributions were determined from the graph the geometric mean as the 50% point (median) and the logarithmic standard deviation as the ratio of the diameters at the 84 and 50% points. 

Plot the summation curve of the data given in Problem 1 on log-probability grid and obtain the logarithmic standard deviation. What is the value of the uniformity modulus ... 

By methods explained in Chapter 3, this equation for a distribution having a known geometric mean dg and logarithmic standard deviation [Pg.328]

Determine the fracture probability of structures and components. This is rather conventionally undertaken using fragility curves which relate the conditioned probability of fracture with the maximum acceleration of the component/struc-ture. The simplification introduced by the fragility curves consists in the fact that they are supposed to depend on three parameters only a median rupture acceleration, /, and two logarithmic standard deviations (log-normal distribution), Par and Pav, related to the intrinsic variability of the component behaviour and to the variability... 

HCLPp. A value for the selected parameter representing an external event, chosen on a median capacity curve of a structure, system or component, with and random variables with unit medians, representing the inherent randomness of the median and the uncertainty in the median value respectively. Assuming that both and % are log-normally distributed with logarithmic standard deviations and pu, respectively, the variables A, and determine a family of fragility curves representing various levels of confidence. The point on the 95% confidence curve that corresponds to a 5% Pp is commonly referred to as the high confidence of a low probability of failure (HCLPp) value, therefore ... 

Pd = logarithmic standard deviation of the uncertain capacity of the asset to resist damage state d. 

Information-theory reasons. Given a median and logarithmic standard deviation of some uncertain positively valued quantity like component capacity, the lognormal is the most uncertain distribution, that is, it assumes the least amount of information. 

Bounding-Failure Excitation Suppose one possesses observations where at least one specimen did not fail, at least one specimen did fail, and one knows the peak excitation to which each specimen was subjected, but not the actual excitation at which each specimen failed. These data are referred to here as bounding, or type B, data. Specimens are grouped by the maximum level of excitation to which each specimen was subjected. Assume the fragility function is reasonably like a lognormal cumulative distribution function and find the parameter values 6 (median) and jS (logarithmic standard deviation) as follows ... 

P(s) = logarithmic standard deviation of the vulnerability function, i.e., the standard deviation of the natural logarithm of the damage factor when the asset is exposed to excitation s. 

For example, assume that loss is lognormally distributed conditioned on shaking s, with median 6 s) and logarithmic standard deviation jS(s) as described near Eq. 42, which are related to the mean vulnerability function y(s) and coefficient of variation v(i) as in Eqs. 6 and 7. Under the assumption of Poisson arrives of earthquakes, shaking with 10 % exceedance probability in 50 years is the shaking with exceedance rate G spml) = 0.00211 per year, so PML can be estimated as a fraction of value exposed by... 

Conditional Spectra, Fig. 5 Intensity-based assessments of peak story drift ratio (median and logarithmic standard deviation in (a) and (b) respectively) of a... 

In the second step of the fragility analysis, the assumption that = 1 is relaxed. Instead, [/ is assumed to be log-normally distributed with logarithmic standard deviation jSy. 

The individual safety factors F, are assumed to follow a log-normal distribution. Their chara ter-istic parameters are their median values F, -representing the (often conservative) bias - and the logarithmic standard deviations and representing the variability due to aleatory and epistemic nondeterminism, respectively. The most widely used method for estimating these characteristic parameters is the approximate second moment procedure. 

In order to estimate the logarithmic standard deviations fii R and fSi u, it is necessary to perform an additional calculation of the safety factor, in which all model variables are median centered, with the exception of the input variables corresponding to the analyzed partial safety factor. The latter ones are instead perturbed by a multiple k of their standard deviation. The input variables should be perturbed to the side which leads to a smaller safety factor (since we are more interested in quantifying the effect of variables in the more unfavorable cases). The resulting partial safety factor is denoted here as. ... 

Finally, the logarithmic standard deviations of the individual safety factors are combined using square root of sum of squares (SRSS), in order to obtain the corresponding variabihty parameter of the global safety factor, Pr and jSy, respectively. 

The functional form most commonly used for components fragility functions is the lognormal distribution that asks for the definition of median and logarithmic standard deviation. Therefore, for each component, it is necessary to define median capacity and logarithmic standard deviation (CTiog) in terms of EDP for each damage state. 

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