Gamma distribution moments

The mean of the gamma distribution is cufl and its variance a/ 2. The moment generating function is... 

More terms of the series are usually not justifiable because the higher moments cannot be evaluated with sufficient accuracy from experimental data. A comparison of the fourth-order GC with other distributions is shown in Fig. 23-12, along with calculated segregated conversions of a first-order reaction. In this case, the GC is the best fit to the original. At large variances the finite value of the ordinate at fr = 0 appears to be a fatal objection to both the Gaussian and GC distributions. On the whole, the gamma distribution is perhaps the best representation of experimental RTDs. 

As simple as these equations are, they result in some unexpected problems. Consider first the case where is assigned. The practical problem becomes the determination of the bubble point at some given temperature. The pressure corresponding to the bubble point is —that part is easy. One can now renormalize to x = bIp, and use Eq. (33) to calculate the vapor phase composition X (x). Notice, however, that X (a ) will not have the same functional form as X (x) while the latter has a first moment equal to unity, = 1, the former has (- l)th moment equal to unity, = 1. If, for instance, one uses a gamma distribution,. J (x) = a,x), (x) will not be given by a gamma distribution. 

The parameter , is limited to integer values. Calculations involving expectations of AL) are greatly simplified by the choice of = 2 that provides a balance between flexibility and tractability ( = 2 was used in this study). The moment-generating function of the gamma distribution (Rice, 1995) is used to obtain expressions for the mean m(L) and variance v(L) of L given as ... 

We now obtain an approximate solution by the method of moments (see Section 11.1), using the parametric method and assuming that the initial distribution is a gamma - distribution,... 

To close the system of equations represented by Eq. (15.41) it is necessary to express the right-hand part in terms of moments. To this end, the coagulation kernel should have a special form (for example, the form of a homogeneous polynomial of degrees V and co), or it is necessary to accept that the distribution conforms to a certain class (for example, a logarithmic normal distribution or a gamma distribution). The first method is called the method of fractional moments, and the second one the parametric method. 

In the parametric method, it is assumed that the distribution with time remains as a gamma distribution and that the parameter Vb changes with time. If only drop coagulation is taken into account, there is constancy of the volume concentration W, i.e. mi. When limited to a two-moment approximation, one obtains Eqs. (11.30) for n( V, t), Vb, and the numerical concentration of drops N. Substitution of Eqs. (15.40) and (15.30) into these equations and elimination of Vb leads to an ordinary differential equation for N(t) ... 

The set of moment expressions to be considered consists of either the live and dead moments [Eqs. (70)-(75)j or the live and bulk moments [Eqs. (70), (71) and (76)-(78)], substituting and H2 for Ci and C2 in Eq. (71). Choice of the former, while it is common practice in the literature, suffers from the problem that the equations for X2 and C2 depend on C3. The Hulburt and Kutz [88] method is often used to approximate C3, assuming that the molecular weight distribution can be represented by a truncated series of Laguerre polynomials by using a gamma distribution weighting function [Eq. (79)]. ... 

Several of the standard statistical distributions are described by Hahn Shapiro (Statistical Models in Engineering, 1967) with mention of their applicability. The most useful models are the Gamma (or Erlang) and the Gaussian and some of their minor modifications. As an illustration of something different the Weibull distribution is touched on in problem P5.02.18. These distributions usually are representable by only a few parameters that define the asymmetry, the peak and the shape in the vicinity of the peak. The moments are such parameters. 

Likewise, the distribution of the sum Z of two gamma variables X and Y with identical second parameter ft and with moment generating functions Mx(t)=( 1 —fk) ax and My(t)=(l—f t) ar has interesting additive properties. Again... 

The most familiar estimation procedure is to assume that the population mean and variance are equal to the sample mean and variance. More generally, the method of moments (MOM) approach is to equate sample moments (mean, variance, skewness, and kurtosis) to the corresponding population. Software such as Crystal Ball (Oracle Corporation, Redwood Shores, CA) uses MOM to fit the gamma and beta distributions (see also Johnson et al. 1994). Use of higher moments is exemplified by fitting of the... 

This equation with a fixed value of N and a variable value of t is not a Poisson distribution function (in contrast to Eq. 6.3), but a gamma density function [14], with a first moment given by... 

Use of Eq. [23] automatically implies a power law distribution of pore sizes. A variety of other distribution functions, including the log-normal, incomplete Gamma, and Weibull distribution functions, have also been used to characterize natural pore-size distributions (e.g., Brutsaert, 1966). Furthermore, it is possible to parameterize the pore size distribution without resorting to a particular distribution function model using moment analysis (Brutsaert, 1966 Powers et al., 1992). 

For the definition of the complete and incomplete gamma functions, F(a) and P a,x), see [28, Chap. 6]. Thus Rpa is the radius of a homogeneous charge density distribution yielding the same value for the Barrett moment as the charge density distribution under discussion. For Barrett equivalent radii parameters see, e.g., [35,46,47]. 

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