Multinomial expansion

Fig. 2.17 Portion of a pulsed-laser time-of-flight spectrum showing the formation of doubly charged Mo diatomic cluster ions in pulsed-laser stimulated field evaporation of Mo. From a multinomial expansion analysis, it is concluded that few Mo+ ions are formed. Although there are a total of only 438 ions in the entire spectrum, all the expected 15 mass lines due to an isotope mixing of the 7 Mo isotopes are present. A mass line, even when it contains only three ions such as that of Min = 100, is clearly identifiable.

Mo+ and Mc>2+ one may resort to a multinomial expansion coefficient analysis. Let us represent the fractional abundances of the 7 Mo isotopes by a, b, c, d, e, f and g. The calculated and experimental fractional abundances of the 15 mass lines in Mo + are listed in Table 2.4. When the calculated abundances are compared with the experimental abundances, one reaches the conclusion that the spectrum shown in Fig. 2.17 contains few if any Mo+ ions. If on the other hand, a fraction of the ions are Mo+, say p, then the relative abundances of each mass line can also be calculated. For example, that of Min = 92 should be [pa + (1 — p)a2], and that of M/n = 94 should be [pb + (1 — p)(b2 + 2ad), etc. Thus the fraction p can be obtained by best fit of theoretical abundances and experimental abundances of different Min mass lines. 

The total number of coins C (20 in this case) serves as the constraint. The number of ways each distribution can be generated, which we will call Q, turns out to be given by what is called the multinomial expansion ... 

Just as with the binomial distribution, calculating factorials is tedious for large N. The binomial distribution converged to a Gaussian for large N (Equation 4.11). The most probable distribution for the multinomial expansion converges to an exponential ... 

One immediately realizes that the previous expression can be rewritten more compactly using the multinomial expansion [17]... 

Note that the sums on the right-hand sides can be rewritten as multinomial expansions. We will make use of this fact at several points in the derivation. 

If the multinomial expansion of the potential is abandoned, then in both the VT and PT approaches a major bottleneck is the evaluation of matrix elements. This severely limits the applicability of these methods to molecules beyond tetraatomics. 

Butler, Durbin, and Helvacian (1996) use this distinction between diffieult-to-monitor and easy-to-monitor injuries to explore whether soft-tissue injury elaims correlate with level of benefits and spread of HMOs. They find in their 10-year, 15-state sample of workers compensation claims that the proportion of claims attributable to soft-tissue injuries rose from 44.7 percent of all claims in 1980 to 50.6 percent in 1989. Concurrently, the share of costs attributable to soft-tissue injuries rose from 41 pereent to 48.8 percent. The share of costs for injuries that crush or fracture a bone—easy-to-monitor claims—is the only category that declined between 1980 and 1989. Using a multinomial logit model, the authors determine that most of the increase in soft-tissue injury is attributable to the expansion of HMOs. Specifically, they ascribe the rise in such injuries to moral hazard response by HMO providers, who increase their revenue by classifying as woik-related injuries as many health conditions as possible. ... 

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