Distribution Normal mathematical

Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution cur ves. It is most unlikely that any frequency distribution cur ve obtained in practice will exactly fit a cur ve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. 

Of particular interest in Statistics is that the means of many large samples taken from a particular population are approximately distributed in this Normal fashion, i.e., they are said to be Normally distributed. This is true even when the population data themselves are not Normally distributed. The mathematical properties of a true Normal distribution allow quantitative statements of the area under the curve between any two points on the x-axis. In Section 6.6.1 it was shown that the total area under the Normal curve is 1, or 100%. It is also of interest to know the proportion of the total area under the curve that lies between two points that are equidistant from the mean. These points are typically represented by multiples of the SD. From the properties of the mathematical equation that governs the shape of the Normal curve, it can be shown that ... 

Probability distribution. A mathematical model which, for a range of values, associates a probability (or probability density) with a given value. Common examples are the Normal binomial, t, F, chi-square, exponential (or negative exponential) and Poisson distributions. 

The Gaussian distribution is the best known distribution. It represents the error law about the arithmetic mean. Because of its frequent appearance, the Gaussian distribution is also called the normal distribution in mathematics. In contrast, a certain form of the Schulz-Flory distribution is often called the normal distribution in macromolecular science. 

The basis of the approach rests on Gosset s proof that, for a theoretical normal distribution of a quantity x with standard deviation a, the means of random sanples of n independent measurements of x are also normally distributed but with standard deviation a/n, i.e., the standard deviation of the mean, or standard error SE (Equation [8.5]). Moreover, if it turns out that the tme distribution is not tmly Gaussian, i.e., is not well described by Equation [8.3], the distribution of the mean values of several small sub-sets does tend to be well approximated by its own normal distribution. The mathematical theory underlying the Student s-t approach is well outside the purposes of the present book and we shall only discuss briefly the aspects that are of direct importance for its judicious application. 

Because the probability distribution is based on the number fraction, the average is called the niunber average molecular weight. It is the normal mathematical definition of an average. 

A particle size distribution having a symmetrical bell shape about the mean. Sometimes known as the normal or normal probability distribution. The mathematics for the curve were developed by the mathematician De Moivre, but also associated with it were, otlier great mathematicians. Gauss and Laplace... 

ABSTRACT The paper presents a probabilistic method to assess lifetimes of devices/components that operate under conditions typical of ageing processes. It has been assumed that the random rate of the component s wear is of the form taken by the failure rate function for the Weibull distribution, or approximately follows the linear pattern. From the point of view of mathematics, it has been based on the difference equation that, after some rearrangements, results in a partial differential equation of the Fokker-Planck type. From the particular solution to this equation one gets density function of the wear-and-tear in the form of normal distribution. Having found the density function of the wear-and-tear, one can formulate a relationship for reliability for the assumed permissible value of the wear-and-tear. With the normal distribution normalized and the required level of reliabUity reached, one can then compute the lifetime of a device or component under consideration. 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very specif properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

Both func tions are tabulated in mathematical handbooks (Ref. I). The function P gives the goodness of fit. Call %q the value of at the minimum. Then P > O.I represents a believable fit if ( > 0.001, it might be an acceptable fit smaller values of Q indicate the model may be in error (or the <7 are really larger.) A typical value of for a moderately good fit is X" - V. Asymptotic Iy for large V, the statistic X becomes normally distributed with a mean V ana a standard deviation V( (Ref. 231). 

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

The numerator is a random normally distributed variable whose precision may be estimated as V(N) the percent of its error is f (N)/N = f (N). For example, if a certain type of component has had 100 failures, there is a 10% error in the estimated failure rate if there is no uncertainty in the denominator. Estimating the error bounds by this method has two weaknesses 1) the approximate mathematics, and the case of no failures, for which the estimated probability is zero which is absurd. A better way is to use the chi-squared estimator (equation 2,5.3.1) for failure per time or the F-number estimator (equation 2.5.3.2) for failure per demand. (See Lambda Chapter 12 ),... 

The principle of this technique is based on the fact that small particles scatter light further than do larger ones. Therefore, though the mathematics is complex and normally incorporated within vendors software, by determining the light intensity at positions in the focal plane, the size distribution can be inferred. 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The distribution of the vectors normal to the surface is particularly interesting since it can be obtained experimentally. The nuclear magnetic resonance (NMR) bandshape problem, for polymerized surfaces, can be transformed into the mathematical problem of finding the distribution function f x) of... 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

Mathematical calculations of the current-density distribution in a direction normal to the electrode are rather difficult hence, to discuss the major qualitative trends, we shall limit ourselves to reviewing the simplest cases. Consider the processes occurring in a porous electrode of thickness d operated unilaterally. The current density generated at depth x per unit volume will be designated as and it is obvious that iyj X ... 

The method has several advantages, the first being its mathematical simplicity, which does not prevent it from yielding classification results as good and often better than the much more complex methlods discussed in other sections of this chapter. Moreover, it is free from statistical assumptions, such as normality of the distribution of the variables. 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

If the mathematical model represents adequately the physical system, the error term in Equation 2.3 represents only measurement errors. As such, it can often be assumed to be normally distributed with zero mean (assuming there is no bias present in the measurement). In real life the vector e, incorporates not only the experimental error but also any inaccuracy of the mathematical model. 

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form,... 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

Several mathematical models are available for predicting the dissolution of particles of mixed size. Some are more complex than others and require lengthy calculations. The size of polydisperse drug particles can be represented with a distribution function. During the milling of solids, the distribution of particle sizes most often results in a log-normal distribution. A log-normal distribution is positively skewed such that there can exist a significant tail on the distribution, hence a number of large particles. The basic equation commonly used to describe the particle distribution is the log-normal function,... 

The desorption isotherm approach is the second generally accepted method for determining the distribution of pore sizes. In principle either a desorption or adsorption isotherm would suffice but, in practice, the desorption isotherm is much more widely used when hysteresis effects are observed. The basis of this approach is the fact that capillary condensation occurs in narrow pores at pressures less than the saturation vapor pressure of the adsorbate. The smaller the radius of the capillary, the greater is the lowering of the vapor pressure. Hence, in very small pores, vapor will condense to liquid at pressures considerably below the normal vapor pressure. Mathematical details of the analysis have been presented by Cranston and Inkley (16) and need not concern us here. 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

Of highest practical relevance is the case in which the scattering pattern, the structural entities, and even the orientation distribution g(uniaxial symmetry (F3-materials). If the structure is ruled by polydispersity and the material is uniaxially oriented, F3 is most frequently fulfilled. In this case the mathematical relations are considerably simplified. Suitably the orientation distribution is normal-ized... 

Under the proper conditions (said conditions being that the errors that prevent all the data relationships from being described by a single equation are normally [1, 2] distributed) it can be proven mathematically that the most probable equation is exactly the one that is the least square equation. While we have discussed this point... 

Here again, our task is simplified by the two facts we have mentioned above first, that we can reuse many of the results we obtained previously for the case of Normally distributed noise, and second, that the nature of uniformly distributed noise characteristics simplify the mathematical analysis. Our first step in this analysis starts with equation 44-71, that we derived previously in Chapter 44 referenced as [5] as a general description of noise behavior ... 

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