Lognormal distribution properties

The hypothesis to be tested requires an appropriate test statistic. Since acute toxins are being considered here, it is essential to choose a statistical measure that is likely to identify lognormal distributions that potentially produce large values, even if these values are improbable. A sensitive statistic tic must combine both the overall level (mean) and the intrinsic variability (variance). A test statistic with this property is the estimated 95th percentile defined as... 

The modal approach assumes a shape for the particle-size distribution, typically one or more lognormal distributions, and represents evolution of the size distribution as evolution of the parameters characterizing the distribution, i.e., the amplitude, mode radius, and variance for the lognormal distribution (Binkowski and Shankar, 1995 Wilck and Stratmann, 1997). This approach offers the possibility of representing aerosol microphysical properties in models with far fewer variables (modal parameters) than are required in the sectional method (numbers of particles in each bin). 

Kosugi, K. 1996. Lognormal distribution model for unsaturated soil hydraulic properties. Water Resour. Res, 32 2697-2703. 

We have discussed the properties of the lognormal distribution for the number concentration. The next step is examination of the surface and volume distributions corresponding to a lognormal number distribution given by (8.34). Since ns(Dp) = nD nN(Dp) and nv(Dp) = (n/6)DpriN(Dp), let us determine the forms of ns(Dp) and nv(Dp) when n(Dp) is lognormal. From (8.34) one gets... 

The PSA approach to the evaluation of probabilistic pressure capacity involves limit state analyses. The limit states represent the ensemble of possible failure modes of the confinement functions. In principle, containment failure may be interpreted as incipient leakage (which may be a small controlled le ) or a large catastrophic break. The median capacity of a given failure mode is typically dependent on several factors, including materid properties, modelling assumption and failure criteria. Considerable uncertainty and variability may be introduced in these factors and should be properly accounted for in the probabilistic overpressure fragility evaluation. To incorporate the uncertainty and variability in the formulation, the pressure capacity for each failure mode is represented as a random variable with a lognormal distribution. 

After the selection of the analytical model, the next step is to generate the structural simulations. Due to the probabilistic nature of seismic fragility analysis, some of the major structural parameters within the analytical model are considered as random variables with appropriate probability density functions assigned to them. Normal or lognormal distributions are commonly used for convenience. These can be mechanical properties like stiffness or strength to account for the material variability or geometric properties like... 

The preceding sections have focused on the properties of distributions in general without considering any particular type of distribution. In this section, we describe the characteristics and applications of the lognormal distribution for aerosol particle size analysis. As discussed below, the normal distribution, although widely used elsewhere, is not suitable for most aerosol particle size distributions. 

All moment distributions of any lognormal distribution will be lognormal and have the same geometric standard deviation. This means that they will have the same shape when they are plotted on a logarithmic scale. The lognormal distribution is the only common distribution that has this property. Figure 4.10 shows the distri-... 

This value falls below the theoretical ratio for these isotopes but is based on the same efficiency and half-life data which were used to derive the atoms/Mg relationships. Therefore, we will find fa and fa so that the ratio of the total 137Cs to the total 155Eu equals 8.14. It is a property of the lognormal form of distribution function that if particle mass distribution is described by x and a, then the surface/volume distribution is given by (4) ... 

The continuously compounded rate of return is an important component of option pricing theory. If r is the continuously compounded rate of return, we can use the lognormal property to determine the distribution that this follows. At a future date T, the asset price S may be written as Equation (2.21) ... 

Using the lognormal property, we can describe the distribution of the risk-free rate as ... 

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