Binomial Approach

The above stepwise treatment of X+1, X+2 and X+3 peaks has the advantage that it can be followed easier, but it bears the disadvantage that an equation needs to be solved for each individual peak. Alternatively, one can calculate the relative abundances of the isotopic species for a di-isotopic element from a binomial expression. [2,13,14] In the term a + b) the isotopic abundances of both isotopes are given as a and b, respectively, and n is the number of this species in the molecule. 

For n = 1 the isotopic distribution can of course be directly obtained from the isotopic abundance table (Table 3.1 and Fig. 3.1) and in case of n = 2, 3 or 4 the expression can easily be solved by simple multiplication, e.g.. 

we obtain w+1 terms for the isotopic pattern of w atoms. The binomial approach works for any di-isotopic element, regardless of whether it belongs to X+1, X+2 or X-1 type. However, as the number of atoms increases above 4 it is also no longer suitable for manual calculations. 

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The halogens Cl and Br occur in two isotopic forms, each of them being of significant abundance, whereas F and I are monoisotopic (Table 3.1). In most cases there are only a few Cl and/or Br atoms contained in a molecule and this predestinates the binomial approach for this purpose. 

The polynomial approach is the logical expansion of the binomial approach. It is useful for the calculation of isotopic distributions of polyisotopic elements or for formulas composed of several non-monoisotopic elements. [2,14] In general, the isotopic distribution of a molecule can be described by a product of polynominals... 

One popular way of turning theory into practice is to use a tree approach to modelling. The tree can be either binomial or trinomial in its construction. To illustrate the idea consider first the binomial approach. The tree could be set up to reflect observed or estimated market short rates and the data provided in Exhibit 18.2 will help to demonstrate this idea. 

As an alternative to this binomial approach, Hull and White have suggested a two-stage methodology that uses a mean-reverting process with the short rate as the source of uncertainty and calculated in a trinomial tree framework. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

Determine the copolymer composition for a styrene-acrylonitrile copolymer made at the azeotrope (62 mol% styrene). Assume = 1000. One approach is to use the Gaussian approximation to the binomial distribution. Another is to synthesize 100,000 or so molecules using a random number generator and to sort them by composition. 

Return now to the binomial distribution [Eq. (1)] and let n approach infinity. The result is then... 

This result is obtained from the binomial distribution if we let p approach 0 and n approach infinity. In this case, the mean fx = p approaches a finite value. The variance of a Poisson distribution is given as cr = fx. 

If indeed the basic goal of equilibrium sampling is to estimate state populations, then these populations can act as the fundamental observables amenable to the types of analyses already described. In practical terms, following 10, a binomial description of any given state permits the effective sample size to be estimated from the populations of the state recorded in independent simulations — or from effectively independent segments of a sufficiently long trajectory. This approach will be described shortly in a publication. 

It has already been mentioned that certain distributions can be approximated to a normal one. As the size of the sample increases, the binomial distribution asymptotically approaches a normal distribution. This is a useful approximation for large samples. 

The more particles the samples contain, the narrower the distribution. In samples taken from true solutions, where the ultimate particles are molecules, the number of molecules in the smallest practical withdrawn sample is enormous, the variance will approach a value of zero, and the distribution will be virtually uniform. Figure 7.35 demonstrates the effect of the sample size on the shape of the binomial distribution. 

This probability distribution is called the (equal probability) binomial distribution, and is the same distribution that is obtained for tossing an unbiased coin. In books on statistics,19 it is shown that for large values of N, the binomial distribution approaches the continuous normal distribution ... 

Such a population will approach equilibrium where F = (1 — a)/(l + ) (2). The parameter a defines the binomial frequency for cross-fertilization, and (1 — a) defines the frequency for self-fertilization. The population can now be described in terms of its reproductive mode (3)—viz.,... 

Figure 8.3 Convergence with number of binomial moments of hydration free energy predicted using several default models for a spherical solute with distance of closest approach 3.0 A for water oxygen atoms. Identifications are diamonds (dash-dot lines), hard-sphere default HS), crosses (short dash line), Lennard-Jones LJ) default squares (long dash line), Poisson default triangles (dotted line), cluster Poisson default and circles (solid line), flat default. For this circumstance, yth-order binomial moments are non-zero through j = 9, and the horizontal line is the prediction with all nine moments included. Among the predictions at j = 2, the best default model is the Lennard-Jones case. But with the hard-sphere model excepted, the differences are slight. See Hummer et al. (1996), Gomez et al. (1999) and Pratt etal. (1999) for details of the calculations.

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