Ratio of First and Zeroth Moments

In general terms, the ratio of the rst moment to the zeroth moment of any distribution gives the arithmetic mean. For an unnormahzed number distribution, the zeroth (j = 0) and rst j = 1) moments of the distribution about zero are given, respectively, by [cf. Eq. (P4.1.1)]  

The ratio of these moments is the arithmetic mean of the number distribution and by comparing with Eq. (4.5) we can write 

The arithmetic mean of an unnormalized weight distribution is likewise given by [cf. Eq. (4.10)]  

In the above two examples, we have chosen unnormalized distributions. For normalized distributions, the area under the curve for the differential number distribution [Fig. 4.1(a)] or weight distribution [Fig. 4.2(a)] equals unity. That is. 

It is now seen from Eqs. (4.17) and (4.18) that the arithmetic mean is numerically equal to the rst moment of the normalized distribution. 

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If one now chooses the appropriate induced dipole model, Eqs. 4.1 through 4.3, or a suitable combination of these, with N parameters po, >7, R0,. .., and one has at least N theoretical moment expressions available, an empirical dipole moment may be obtained which satisfies the conditions exactly, or in a least-mean-squares fashion [317, 38]. We note that a formula was given elsewhere that permits the determination of the range parameter, 1/a, directly from a ratio of first and zeroth moments it was used to determine a number of range parameters from a wide selection of measured moments [189]. In early work, an empirical relationship between the range parameter and the root, a, of the potential is assumed, like 1/a 0.11 a. That relationship is, however, generally not consistent with recent data believed to-be reliable. 

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