Second moment definition

The numerical value of the exponent k determines which moment we are defining, and we speak of these as moments about the value chosen for M. Thus the mean is the first moment of the distribution about the origin (M = 0) and is the second moment about the mean (M = M). The statistical definition of moment is analogous to the definition of this quantity in physics. When Mj = 0, Eq. (1.11) defines the average value of M this result was already used in writing Eq. (1.6) with k = 2. 

From this definition, we see it is the ratio of the third moment of the distribution about the molecular weight origin to the second moment about the origin. 

There are several different definitions relating to second moments in the literature, and it is necessary to take care when applying seemingly simple formulae that involve such quantities. Not only that, authors often use the same name for the different definitions. 

It is curious that the chair- boat problem, which is most associated with small, liquid-state molecules, arises in the context of solid-state research (B3, II). Although the paucity of useful experiments militates against a definitive solution here E3), the frequency independence of the NMR second moment (E2), the absence of an observable free-induc-tion decay (Tj <25 fis) in the pulsed NMR spectrum (El), and the smoothness of the absorption mode itself (SI), all argue against the... 

This involves obtaining the mean-residence time, 0, and the variance, (t, of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform,... 

Moreover, using the definition of the second moment of a non-reacting scalar ... 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

If we assume above / = 1 excess fluorine atoms over those required to fill the available surface are in a bulk solid phase of definite composition, then the second moment of the line is the weighted sum of the second moments of fluorine in the bulk phase ((AH ))b and fluorine on the filled surface That is, if n, is the mole fraction on the filled surface and... 

In the nonzero trace definition, this equality is no longer valid. To illustrate the relation for the diagonal elements of the second-moment tensor, we rewrite the xx element as... 

The first and second moments of the multievent distribution f(z n) are denoted z and z, respectively. Because, by definition, D = z, the average number of events n at dose D is ... 

Elongation, by definition, is X = Z/7o. These specimens had been drawn at room temperature to elongations X of 1.0, 2.0, 3.0, and 4.0 (strains of 0, 100, 200, and 300%). At 345°K, the main contribution to the second moment is from the XL phase while at 158°K it is from the NXL phase. It is observed that the second moment increases for more highly elongated specimens, i.e., the line broadens as the NXL molecules are oriented and are thereby restricted in mobility. 

This I(S) is always defined when there is at least one distribution for which. S is true but it need not be finite. Thus, if the domain of definition of. S is the set of probability densities p(x) on the whole x axis, a trivial 5 (i.e., true for every p(x)) and also an S which merely gives the value of the first moment, has I(S) = — oo. On the other hand, if S states that the second moment is close to zero, /( S ) is very large, and I(S) -> + oo as this moment approaches zero. Of course there is no p(x) having a zero second moment (only a point distribution, which is not a p(x)). Thus it might seem natural to define I(S) as + oo when S defines an empty set. Then every S without exception has a unique I(S). 

For simplicity, we assume that this and all the other trajectories of the Gibbs ensemble have the same initial condition, x = 0. We assume that the fluctuation does not have any bias, (i (f)) = 0, and that is stationary and gaussian, according to the definition of Eq. (140). Let us evaluate the second moment of the diffusion process. This means making first the square of Eq. (157) and then averaging over the Gibbs ensemble... 

Numerical values of Aa i and Aai2 are obtained from integration of the experimental measurements using the definition of the first and second moments. 

Similarly, the definition of each diagonal element of the orbital second-moment tensor is formally identical to that of the statistical central second moment., wi2 ... 

Using the definition of the number of stagesN (Equation 2.37) and the equations for the first and second moments (Equations 6.138 and 6.139) one obtains for the transport-dispersive model... 

With this definition (and neglecting charge transfer), the first moment of the distribution is zero. The second moment describes the width of the density of states. The third and fourth moments describe the degree of skewness about the center of the distribution and the tendency to form a gap in the middle of the distribution, respectively. 

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