Cumulative lognormal distributions

Figure 3. Cumulated lognormal distribution determined with point, moment and graphic estimator.

If, for example, F(s) is taken as a cumulate lognormal distribution function, dF(s)lds is the lognormal probability density function, i.e.,... 

The fragility curve is defined by a median value of the engineering demand parameter (EDP), which is, for example, spectral displacement in HAZUS and maximum interstory drift ratio (IDR) in Jayaram et al. 2012, and by the variability associated with the estimation of that damage state for that EDP. The median EDP corresponds to the threshold of the damage state. The curve is assumed to follow the cumulative lognormal distribution function as shown below ... 

Larsen (18-21) has developed averaging time models for use in analysis and interpretation of air quality data. For urban areas where concentrations for a given averaging time tend to be lognormally distributed, that is, where a plot of the log of concentration versus the cumulative frequency of occurrence on a normal frequency distribution scale is nearly linear,... 

Quantal dose-effect plots. Shaded boxes (and the accompanying bell-shaped curves) indicate the frequency distribution of doses of drug required to produce a specified effect that is, the percentage of animals that required a particular dose to exhibit the effect. The open boxes (and the corresponding colored curves) indicate the cumulative frequency distribution of responses, which are lognormally distributed. 

The cumulative distribution function N Dp) for a lognormally distributed aerosol population is given by (8.39). Defining the normalized cumulative distribution... 

FIGURE 8.8 Cumulative lognormal aerosol number distributions plotted on log-probability paper. The distributions have mean diameter of 1 pm and ag = 2 and 1.5, respectively. 

The stochastic dynamics usually applied in finance literature is generated by lognormal- or close-to-lognormal -distributed random variables. Leip-nik (1991) shows that the series expansion of order M of a Gog) characteristic function in terms of the cumulants diverges for M oo. Hence, the... 

We show that the application of the EE is admissible leading to accurate results, even in the case of lognormal-distributed random variables. This good-natured behavior of the EE, firstly comes from the fact that the volatility t5 ically occurring in bond markets is rather low, generating more close-to-normal -distributed random variables. Secondly, the series expansion of the (log) characteristic function in terms of the cumulants can be practically applied for M lower than a critical order Me. 

Together with the one-to-one mapping between the cumulants and the moments (3.2) we can compute the cumulants of a lognormal-distributed random variable x as follows... 

Then, plugging the cumulants (3.15) of a lognormal-distributed random variable z in the lEE scheme leads to the following series expansion of the probability that z exceeds the strike price K... 

On purpose to determine (estimate) the standard deviation tr = tr we have used the theory of Peck Trap 1990 and their special plotting paper for lognormal distribution, where on abscise are represented the cumulated percent failures and on the ordinate the logarithm of time to failure (columns 5 and 2 of Table 1), from Fig. 2. This could he done by one set of time to failure and cmnulated percent failure values, for a lot of 50 components. 

Volumetric (volume of microbeads in each diameter class) and cumulative size distributions were determined by laser light scattering, with a 2602-LC particle analyzer (Malverin Instruments) according to the lognormal distribution model. The mean diameter and the arithmetic standard deviation were calculated from the cumulative distribution curve [22]. 

Clearly, any probability density function and corresponding cumulative probability distribution could be used to describe the uncertainty in the data. Trapezoidal, normal, lognormal, and so on, are used routinely to describe uncertainty in data. However, for simplicity, the following discussions are confined to triangular distributions. The eight-step method for quantifying uncertainty in profitability analysis is illustrated next. 

The lognormal is ubiquitous in probabilistic seismic hazard analysis (PSHA) and probabilistic seismic risk analysis (PSRA). To understand it, consider first the normal (not the lognormal) distribution. If a quantity X is normally distributed with mean fi and standard deviation scalar value in -00 < X < 00. Its cumulative distribution function (CDF) can be expressed as follows ... 

The cumulative distribution for the lognormal distribution is obtained as described in Section 4.1. The cumulative plot shown in Fig. 4.9 is the same as Fig 4.5, but diameter is plotted on a logarithmic scale. Note that the median size is the same for both figures. 

The geometric standard deviation, being the ratio of two sizes, has no units and must always be greater than or equal to 1.0. The CMD (S0% cumulative size) and the GSD can be determined directly from a log-probability plot and completely define a lognormal distribution. 

Figure 14.2b presents the urban aerosol size distribution as cumulative number and volume distributions on a log-probability graph. (See Section 4.5.) The distributions show significant departures from a lognormal distribution, and the contributions of the individual modes are hidden. Figures 14.2c and 14.2d respectively show particle number and volume per unit log interval on an arithmetic scale ver-... 

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