Means and Moments

Residence time distributions can be described by any of the functions W(t), F(t), or fit). They can also be described using an infinite set of parameters known as moments  

The first moment is the mean of the distribution or the mean residence time. 

t can be found from inert tracer experiments. It can also be found from measurements of the system inventory and throughput since 

Agreement of the t values calculated by these two methods provides a good check on experimental accuracy. Occasionally, Equation (15.13) is used to determine an unknown volume or an unknown density from inert tracer data. 

Roughly speaking, the first moment, F, measures the size of a residence time distribution, while higher moments measure its shape. The ability to characterize shape is enhanced by using moments about the mean  

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Phase Space, Distribution Function, Means and Moments... 

Statistics means and moments are defined in terms of a suitable probability density function (PDF). Therefore, in the present context these statistical measures are expressed in terms of the normalized distribution function, P(r, c,t) = /(r, c,t)/n(r,t), having the important mathematical property of a PDF ... 

The function/( C) may have a very simple form, as is the case for the calculation of the molecular weight from the relative atomic masses. In most cases, however,/( Cj will be very complicated when it comes to describe the structure by quantum mechanical means and the property may be derived directly from the wavefunction for example, the dipole moment may be obtained by applying the dipole operator. 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

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. 

The second moment is taken about the mean and is referred to as the variance or square of the standard deviation defined by... 

Table 2.5-2 Mean Variance and Moment-Generating Functions for Several Distributions ...

The overall procedure of laminate-strength analysis, which simultaneously results in the laminate load-deformation behavior, is shown schematically in Figure 4-36. There, load is taken to mean both forces and moments similarly, deformations are meant to include both strains and curvatures. The analysis is composed of two different approaches that depend on whether any laminae have failed. 

Figure 1. The energy of bcc and hep randoiri alloys and the ])ai tially ordered a phase relative to the energy of the fee phase (a), of the Fe-Co alloy as a function of Co concentration. The corresponding mean magnetic moments are shown in (h). The ASA-LSDA-CPA results are shown as a dashed line for the o ])hase, as a full line for the her ]>hase, as a dot-dashed line for the hep phase, and as a dotted line for the fee phase. The FP-GGA results for pure Fe and Co are shown in (a) by the filled circles (bcc-fcc) and triangles (hep-fee). In (b) experimental mean magnetic moments are shown as open circles (bcc), open scpiares (fee) and open triangles (hep).

The variance is the second moment about the mean and indicates the closeness of values to the mean. It is denoted by (population) or (sample) and is given for a continuous random variable by... 

The Characteristic Function.—The calculation of moments is often quite tedious because of difficulties that may be encountered in evaluating the pertinent integrals or sums. This problem can be simplified quite often by calculation of the so-called characteristic function of the distribution from which, as we shall see, all moments can be derived by means of differentiation. This relationship between the characteristic function and moments is sufficient reason for studying it at this time however, the real significance of the characteristic function will not become apparent until we discuss the central limit theorem in a later section. 

As an example of these techniques, we shall calculate the characteristic function of the gaussian distribution with zero mean and unit variance and then use it to calculate moments. Starting from the definition of the characteristic function, we obtain18 ... 

This result checks with our earlier calculation of the moments of the gaussian distribution, Eq. (3-66). The characteristic function of a gaussian random variable having an arbitrary mean and variance can be calculated either directly or else by means of the method outlined in the next paragraph. 

For normal statistics, the mean and the variance are completely sufficient to characterize the process all the other moments are zero. For standard normal or Gaussian statistics (i.e., normal statistics with zero mean), the variance p,2... 

The quantities AUMC and AUSC can be regarded as the first and second statistical moments of the plasma concentration curve. These two moments have an equivalent in descriptive statistics, where they define the mean and variance, respectively, in the case of a stochastic distribution of frequencies (Section 3.2). From the above considerations it appears that the statistical moment method strongly depends on numerical integration of the plasma concentration curve Cp(r) and its product with t and (r-MRT). Multiplication by t and (r-MRT) tends to amplify the errors in the plasma concentration Cp(r) at larger values of t. As a consequence, the estimation of the statistical moments critically depends on the precision of the measurement process that is used in the determination of the plasma concentration values. This contrasts with compartmental analysis, where the parameters of the model are estimated by means of least squares regression. 

The cumulants [26] are simple functions of the moments of the probability distribution of 5V-.C2 = (V- V))2),C3 = (V- V)f),C4 = ((]/-(]/))4) 3C22,etc. Truncation of the expansion at order two corresponds to a linear-response approximation (see later), and is equivalent to assuming V is Gaussian (with zero moments and cumulants beyond order two). To this order, the mean and width of the distribution determine the free energy to higher orders, the detailed shape of the distribution contributes. 

One can obtain an exact analytic solution to the first Pontryagin equation only in a few simple cases. That is why in practice one is restricted by the calculation of moments of the first passage time of absorbing boundaries, and, in particular, by the mean and the variance of the first passage time. 

In most applications, the moments of principal interest are the mean ( and variance (if2) = ( 2) — ( )2. The most widely used approach for approximating... 

That such a matrix describes a metastable state when H0 Hl can easily be seen and will be discussed in the following section. Since / z is related to the mean magnetic moment and fiD to the mean dipole-dipole energy, this raises the question of the independence of these two invariants of the motion. It may be shown10 by introducing the Fourier transforms of the Iz(j)... 

The mean angular moment (in units of %) is fit = dqjdv and the mean dipole-dipole energy is ED — — dq/dp. In the limit of high temperatures we may expand the exponential and retain only the linear terms, v and /9 may then be considered as the coefficients of the expansion of p in orthogonal operators, since Tr(VmMa) — 0. In this limiting case it then becomes evident that v and / are independent. It is easily shown that the thermodynamical temperature, defined by... 

A two-term power series expression can be derived to handle this case, but it will again fail in cases where the skewness depends on x, but the mean and variance are constant. However, note that the beta PDF can be successfully handled with the two-term form since all higher-order moments depend on the mean and variance. By accounting for the entire shape of the mixture-fraction PDF, (5.318) will be applicable to all forms of the mixture-fraction PDF. 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

Alternatively, one may simply write h(x) as a sum over terms h(q) cos(qx +

random number with the proper second moment of b(q) with zero mean and one random number for each phase

uniformly distributed between 0 and ji, and filter the absolute value of b x) in the same way as described in the previous paragraph. Other methods exist with which to generate self-similar surfaces, such as the midpoint technique, described in Ref. 24. 

The fitted means and standard errors for log-transformed comet tail moments, as well as the percentage of cells exhibiting extensive DNA damage (e.g., cells labeled 3 and 4) are reported in Table 2 [see p. 137]. An adjusted p value indicated no differences existed between cells treated with extracts from exposed filters or with hydrogen peroxide. Cellular responses were significantly different (P < 0.05) between unloaded PMj 5 filter extracts and loaded PM2 5 extracts as well as extracts containing deferoxamine. 

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