Binomial normalization

One convenient consequence of binomial normalization for the RDMs (Eq. (6)) is that when this convention is followed, extensive quantities such as Ap have traces proportional to N, while nonextensive quantities possess traces that scale as some higher power of N (e.g., tiDp N ). Let us define a set of quantities... 

As noted earlier, trA N when binomial normalization is used for the ROMs, while nonextensive terms have traces that scale as higher powers of N. This is certainly a convenient means to recognize terms that are not extensive, but in some sense this trick overlooks the physical picture behind extensivity. 

B. Binomial, Normal, and Poisson Distributions—General Comments... 261... 

The distribution of spot sample compositions of a certain size, taken from a randomly mixed batch of A and B, can be calculated theoretically. The methods of calculation are standard statistical techniques, and several papers have shown how various aspects of these basic ideas can be applied to solids mixing. Most of the calculations and discussion center around three distributions binomial, normal, and Poisson. 

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

Normal Distribution The binomial distribution describes a population whose members have only certain, discrete values. This is the case with the number of atoms in a molecule, which must be an integer number no greater then the number of carbon atoms in the molecule. A molecule, for example, cannot have 2.5 atoms of Other populations are considered continuous, in that members of the population may take on any value. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

Furthermore, when both np and nq are greater than 5, the binomial distribution is closely approximated by the normal distribution, and the probability tables in Appendix lA can be used to determine the location of the solute and its recovery. 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

Also, in many apphcations involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution cur ves. It is most unlikely that any frequency distribution cur ve obtained in practice will exactly fit a cur ve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. 

For description of possible ways of foriuation of char ges we use a binomial spread [2, 3] i.e. a normalized differential function of the spread. 

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

It is tempting to use Eq. (7) to derive Eq. (6), because they have similar forms given the relationship of the F function to the factorial. But the binomial and the beta distribution are not normalized in the same way. The beta is nonnalized over the values of 0, whereas the binomial is nonnalized over the counts, y given n. That is. 

Other important probability distributions include tlie Binomial Distribution, the Polynomial Distribution, tlie Normal Distribution, and the Log-Normal Distribution. 

For np > 5 and n( 1 - p) > 5, an approximation of binomial probabilities is given by the standard normal distribution where z is a standard normal deviate and... 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

Using the normal approximation to a binomial distribution, confidence intervals (CIs) for p(y =j X) can be established for a specific significance level, a ... 

Power Calculator provides sample size programs for various models, including normal, exponential, binomial, and correlation models http //home. stat.ucla.edu/ calculators/powercalc/... 

If the normal approximation to the binomial distribution is valid (that is, not more than 20% of expected cell counts are less than 5) for drug therapy and symptom of headache, then you can use the Pearson chi-square test to test for a difference in proportions. To get the Pearson chi-square / -value for the preceding 2x2 table, you run SAS code like the following ... 

Fit data to a recognized mathematical distribution (e.g., normal, Poisson, binomial). When appropriate, transform the data (e.g., log 10 transformation). Calculate confidence limits. 

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

If the variation were completely unpredictable, there would be no hope of rational planning to take it into account. Usually, however, although it is not possible to predict that a given occurrence will certainly happen, it is possible to assign a probability for any particular occurrence. If this is done for all possible occurrences, then, in effect, a probability distribution function has been defined. Certain types of such distributions can be derived mathematically to fit special situations. The normal, Poisson, and binomial distributions are frequently encountered in practice. 

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