Distributions log normal

The log-normal distribution is frequently observed in ceramic powder processing. The log-normal distribution is skewed to larger sizes compared to the normal distribution and has no finite probability for sizes less than zero as seen with the normal distribution. It is obtained by replacing x with z = In d in the normal distribution, which gives the following distribution function [18]  

This distribution can be rewritten in terms of size, d, as follows  

If the number distribution follows a log-normal distribution then the surface area and the weight distributions also follow log-normal distributions with the same geometric standard deviation. Conversion from one log-normal distribution to another is easily done using the following equations for the various mean sizes [18]  

FIGURE 2.11 Linearized Ic normal distribution plot Cumulative % less than 

To transform from the geometric mass mean, to other mean sizes the following equations are used [18]  

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The only two distributions we shall consider are the Gaussian distribution ( normal law ) and the log-normal distribution. 

To obtain the expression for the log-normal distribution it is only necessary to substitute for I and a in Equation (1.52) the logarithms of these quantities. One thus obtains... 

The particle sizes of fillers are usually collected and ordered to yield size distributions which are frequendy plotted as cumulative weight percent finer than vs diameter, often given as esd, on a log probabiUty graph. In this manner, most unmodified fillers yield a straight-line relationship or log normal distribution. Inspection of the data presented in this manner can yield valuable information about the filler. The coarseness of a filler is often quantified as the esd at the 99.9% finer-than value. Deviations from linearity at the high and low ends of the plot suggest that either fractionation has occurred to remove coarse or fine particles or the data are suspect in these ranges. 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

There are many complications with interpreting MWCO data. First, UF membranes have a distribution of pore sizes. In spite of decades of effort to narrow the distribution, most commercial membranes are not notably sharp. What little is known about pore-size distribution in commercial UF membranes fits the Poisson distribution or log-normal distribution. Some pore-size distributions may be polydisperse. 

The graphite microstmcture is assumed to contain a log-normal distribution of pores. Under these circumstances, for a specific defect, the probability that its length falls between c and c+dc is f(c)dc, with f(c) defined as ... 

This means that the average permeability for this heterogeneous medium is the area-weighted average of the average permeability of each of the elements. If the permeability of each element is log-normally distributed, these are the geometric means. 

The size of inhaled particles varies markedly. The size distribution approximates a log-normal distribution that can be described by the median or the geometric mean, and by the geometric standard deviation. For fibers, both... 

A nonnegative random variable X has a log-nonnal distribution whenever In X, the natural logariilun of X, has a normal distribution. The pdf of a random variable x liaving a log-normal distribution is specified by... 

The mean and variance of a random variable X having a log-normal distribution are given by... 

Figure 20.5.3 plots tlie pdf of the log-noniial distribution for a = 0 and (3=1. Probabilities concerning random variables liaving a log-normal distribution can be calculated using tables of the normal distribution. If X lias a log-normal distribution witli parameters a and p, then In X lias a normal distribution with p = a and o = p. Probabilities concerning X can tlierefore be converted into equivalent probabilities concerning In X. Suppose, for example, tliat X lias a log-nonnal distribution with a = 2 and p = 0.1. Then... 

Estimates of the parameters a and p in tlie pdf of a random variable X having a log-normal distribution can be obtained from a sample of observations on X by making use of tlie fact diat In X is normally distributed with mean a and standard deviation p. Tlierefore, tlie mean and standard deviation of the natural logaritluns of tlie sample observations on X furnish estimates of a and p. To illustrate tlie procedure, suppose the time to failure T, in thousands of hours, was observed for a sample of 5 electric motors. The observed values of T were 8, 11, 16, 22, and 34. The natural logs of these observations are 2.08, 2.40, 2.77, 3.09, and 3.53. Assuming tliat T has a log-normal distribution, the estimates of the parameters a and p in the pdf are obtained from the mean and standard deviation of the natural logs of tlie observations on T. Applying the Eqs. (19.10.1), and (19.10.2) yields 2.77 as tlie estimate of a and 0.57 as tlie estimate ofp. 

Other important probability distributions include tlie Binomial Distribution, the Polynomial Distribution, tlie Normal Distribution, and the Log-Normal Distribution. 

Tlie failure rate per year, Y, of a coolant recycle pump has a log-normal distribution. If In Y luis mean, -2, and variance, 1.5, find P(0.175 < Y< 1). Tliree light bulbs (A, B, C) are coiuiected in series. Assume tliat tlie bulb lifetimes are noniially distributed, witli tlie following means and standard deviations. 

Bayes tlicorem and tlie log-normal distribution are used in the first case study (Section 21.2) to obtain an estimate of the failure rate of a coolant recycle pump. 

Measures of potency are log normally distributed. Only p-scale values (i.e., pEC50) should be used for statistical tests. 

Log normal distribution, the distribution of a sample that is normal only when plotted on a logarithmic scale. The most prevalent cases in pharmacology refer to drug potencies (agonist and/or antagonist) that are estimated from semilogarithmic dose-response curves. All parametric statistical tests on these must be performed on their logarithmic counterparts, specifically their expression as a value on the p scale (-log values) see Chapter 1.11.2. 

Irani and Callis (Ref 14) used two parameters of the distribution of ground monocalcium phosphate (which follows the commonly used log normal distribution law) namely, Mg and Og, the geometric mean diameter and the geometric standard deviation, to evaluate the precision and accuracy of electro-formed sieves vs sedimentation as a reference procedure ... 

When n < 0.7, the ln[—ln(l — a)] against In t plots show curvature and linearity is improved if the latter parameter is replaced by t. This reduces the Weibull distribution to a log-normal distribution. Since both exponential and normal distributions are special cases of the more general gamma distribution, Kolar-Anic and Veljkovic [441] compared the applicability of the Weibull and the gamma distributions. The shape parameter of the latter (e) was shown to depend exclusively on the shape parameter of the former (n). 

Mcllvried and Massoth [484] applied essentially the same approach as Hutchinson et al. [483] to both the contracting volume and diffusion-controlled models with normal and log—normal particle size distributions. They produced generalized plots of a against reduced time r (defined by t = kt/p) for various values of the standard deviation of the distribution, a (log—normal distribution) or the dispersion ratio, a/p (normal distribution with mean particle radius, p). 

Fujita [38] showed that for a log-normal distribution of molecular weights (the usual case for polysaccharides) Mz/Mw = Mw/M . 

Gel filtration chromatography has been extensively used to determine pore-size distributions of polymeric gels (in particle form). These models generally do not consider details of the shape of the pores, but rather they may consider a distribution of effective average pore sizes. Rodbard [326,327] reviews the various models for pore-size distributions. These include the uniform-pore models of Porath, Squire, and Ostrowski discussed earlier, the Gaussian pore distribution and its approximation developed by Ackers and Henn [3,155,156], the log-normal distribution, and the logistic distribution. 

Let us now give some examples of the log-normal distribution. A representation of several types of these distributions is given on the next page as 5.5.11. 

In this diagram, the parameters of the various log normal distributions are... 

A log normal distribution will give a straight line when plotted on this ti pe of paper. This means that the PSD is not limited, i.e.- aU sizes of particles are present from - to +°°. However, if the PSD is growth-limited, it will readily apparent from the graph. Ostwald ripening, a mechanism where large... 

Following this example are distributions that do have limits to the "normal distribution. What this means is that the distributions conform to the limits defined in 5.7.6. Note that in 5.8.1., a straight line is evident. This is the type of distribution usually found as a result of most precipitation processes. But as we shall see, this is not true for the other types of log-normal distributions. 

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