Continuous distribution

The Gaussian/normal is distributed according to equation 2.5-2, where jj is the mean, o is the standard deviation, and x is the parameter of intere.st, e.g., a failure rate. By integrating over the distribution, the probability of x deviating from fi by multiples of a arc given in equations 2.5-3a-c. 

For example, it might be stated that there is a 5% probability of x being more than 2 a from the mean. This is true only if the distribution is known to be normal. However, if the distribution is unknown but not pathological, the Chebyshev inequality provides an estimate. 

more restrictive inequality, is given by equation 2.5-5. For 3 a, it gives 0.95 which is much closer to the correct value of 0.997. 

The Central Limit Theorem gives an a priori reason for why things tend to be normally distributed. It says the sum of a large number of independent random distributions having finite means and variances is normally distributed. Furthermore, the mean of the resulting distribution the sum of the individual means the combined variance is the sum of the individual variance..  

The Reactor Safety Study extensively used the lognormal distribution (equation 2.5-6) to represent the variability in failure rates. If plotted on logarithmic graph paper, the lopnormal distribution is normally distributed. 

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Many companies choose to represent a continuous distribution with discrete values using the p90, p50, plO values. The discrete probabilities which are then attached to these values are then 25%, 50%, 25%, for a normal distribution. 

Such an ensemble of systems can be geometrically represented by a distribution of representative points m the F space (classically a continuous distribution). It is described by an ensemble density fiinction p(p, q, t) such that pip, q, t)S Q is the number of representative points which at time t are within the infinitesimal phase volume element df p df q (denoted by d - D) around the point (p, q) in the F space. 

In Ih e quail tiiin mechanical description of dipole moment, the charge is a continuous distribution that is a I linction of r. and the dipole moment man average over the wave function of the dipole moment operator, p ... 

For a continuous distribution, summation may be replaced by integration and by assuming a Gaussian distribution of size, Stoeckli arrives at a somewhat complicated expression (not given here) which enables the total micropore volume IFo, a structural constant Bq and the spread A of size distribution to be obtained from the isotherm. He suggests that Bq may be related to the radius of gyration of the micropores by the expression... 

The most commonly encountered continuous distribution is the Gaussian, or normal distrihution, where the frequency of occurrence for a value, X, is given by... 

A continuous distribution function is a mathematical function which gives N as a function of M. This is the most general way of describing the... 

Since we have ended up with a continuous distribution function, it is more appropriate to multiply both sides of Eq. (1.34) by dx and to say that the equation gives the probability of x values between x and x + dx for n steps of length 1. 

Aerosol Dynamics. Inclusion of a description of aerosol dynamics within air quaUty models is of primary importance because of the health effects associated with fine particles in the atmosphere, visibiUty deterioration, and the acid deposition problem. Aerosol dynamics differ markedly from gaseous pollutant dynamics in that particles come in a continuous distribution of sizes and can coagulate, evaporate, grow in size by condensation, be formed by nucleation, or be deposited by sedimentation. Furthermore, the species mass concentration alone does not fliUy characterize the aerosol. The particle size distribution, which changes as a function of time, and size-dependent composition determine the fate of particulate air pollutants and their... 

For systems following invariant growth the crystal population density in each size range decays exponentially with the inverse of the product of growth rate and residence time. For a continuous distribution, the population densities of the classified fines and the product crystals must be the same at size Accordingly, the population density for a crystallizer operating with classified-fines removal is given by... 

Heterogeneity Adsorbents and ion exchangers can be physically and chemically heterogeneous. Although exceptions exist, solutes generally compete for the same sites. Models for adsorbent heterogeneity have been developed for both discrete and continuous distributions of energies [Ross and Olivier, On Physical Adsorption, Interscience, New York, 1964 Jaroniec and Madey, Rudzinsld and Everett, gen. refs.]. 

The estimation of the mean and standard deviation using the moment equations as described in Appendix I gives little indication of the degree of fit of the distribution to the set of experimental data. We will next develop the concepts from which any continuous distribution can be modelled to a set of data. This ultimately provides the most suitable way of determining the distributional parameters. 

For mathematieal traetability, the experimental data ean be modelled with a continuous distribution whieh will adequately deseribe the pattern of the data using just a single equation and its related parameters. 

Fig. 2.5 I The discrete distribution of probability of crap scores has impressed on it a continuous distribution with quite a different meaning...

The population balance accounts for the number of particles at each size in a continuous distribution and may be thought of as an extension of the more familiar overall mass balance to that of accounting for individual particles. 

The popuiation baiance provides the mathematicai framework incorporating expressions for the various crystai formation, aggregation and disruption mechanisms to predict the finai particie size distribution. Note, however, that wiiiie particies are commoniy characterized by a iinear dimension the aggregation and particie disruption terms aiso require conservation of particie voiume. It was shown in Chapter 2 that the popuiation baiance accounts for the number of particies at each size in a continuous distribution. The quantity conserved is thus the number (popuiation) density and may be thought of as an extension of the more famiiiar mass baiance. The popuiation baiance is given by (Randoiph and Larson, i988)... 

Obviously, the theory outhned above can be applied to two- and three-dimensional systems. In the case of a two-dimensional system the Fourier transforms of the two-particle function coefficients are carried out by using an algorithm, developed by Lado [85], that preserves orthogonality. A monolayer of adsorbed colloidal particles, having a continuous distribution of diameters, has been investigated by Lado. Specific calculations have been carried out for the system with the Schulz distribution [86]... 

A similar acoustic technique was applied by Pickles and Bittleston (1983) to investigate blast produced by an elongated, or cigar-shaped, cloud. The cloud was modeled as an ellipsoid with an aspect ratio of 10. The explosion was simulated by a continuous distribution of volume sources along the main axis with a strength proportional to the local cross-sectional area of the ellipsoid. The blast produced by such a vapor cloud explosion was shown to be highly directional along the main axis. 

In their classic review on Continuous Distributions of the Solvent , Tomasi and Persico (1994) identify four groups of approaches to dealing with the solvent. First, there are methods based on the elaboration of physical functions this includes approaches based on the virial equation of state and methods based on perturbation theory with particularly simple reference systems. For many years... 

Cole and Davidson s continuous distribution of correlation times [9] has found broad application in the interpretation of relaxation data of viscous liquids and glassy solids. The corresponding spectral density is ... 

Let us return to the problem of what type of continuous distribution to choose for the results of an analytical method. The classical choice... 

Figure 5-5. The quantum mechanical picture discrete population histograms take the place of continuous distributions. The overall paramagnetism increases with increasing field strength.

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