Gram-Charlier distribution

Construct the third and fourth order Gram-Charlier distributions corresponding to an Erlang with cr2(tr) = 0.5 or n = 2. 

Find second order segregated conversion with kC0t = 10. The reactor has a Gram-Charlier distribution with the same first three moments as a third order Erlang distribution. 

Schulz 1982), corresponds to a probability distribution which is the Taylor expansion of Eq. (2.32), and similar to the Gram-Charlier distribution of Eq. 

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... 

Comparisons of Gram-Charlier with data and other distributions are in problems P5.02.15 and P5.02.16. In one of these, the third order GC fits better than the fourth order. More experience is needed, however, before a judgement can be made regarding the relative merits of GC and other distributions. At large variances the finite value of the ordinate at tr - 0 appears to be a fatal objection to both the Gaussian and GC distributions. 

In this problem the Gram-Charlier will be compared with the Gamma Gauss distributions for an impulse response curve with the equation... 

The three-dimensional Gram-Charlier expansion, first applied to thermal motion analysis by Johnson and Levy (1974), is a statistical expansion in terms of the zero and higher derivatives of a normal distribution (Kendal and Stuart 1958). If Dj is the operator d/du], the expansion is defined by... 

As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

The characteristic bell shape of many RTDs can be fit to well-known statistical distributions. Hahn and Shapiro (Statistical Models in Engineering, Wiley, 1967) discuss many of the standard distributions and conditions for their use. The most useful distributions are the gamma (or Erlang) and the gaussian together with its Gram-Charlier extension. These distributions are represented by only a few parameters that can be used to determine, for instance, the mean and the variance. 

From the point of view of mathematics, the zone profiles encountered in chromatography belong to the class of distributions, which can be described in terms of their moments and cumulants [6] by the Gram-Charlier series... 

This expansion describes an arbitary distribution, P , for n>0 as a perturbation about a T-distribution, equivalent to the Gram-Charlier series resulting from a perturbation about a Gaussian distribution using Hermite polynomials [46]. 

Using a Gram-Charlier series with terms up to 4 order for f)((x) and a Gaussian distribution for fy(y) leads to the following set of equations for the stationary state ... 

For one particular set of data In F1g.3 the joint distribution of y and z, according to the method of non-6auss1an closure, Is compared with the distribution obtained by numerical simulation. On one hand It can be observed that the actual distribution of y and z Is entirely different from a joint normal one, hence It can not be approximated by a trunctated Gram-Charlier series in detail. Nevertheless It Is possible to see a strong correlation between both processes and some bimodality. But on the other hand negative values of the probability density function can also be noticed. 

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