The Binomial Distribution

The binomial distribution is a pdf that applies under the following conditions  

The probability that any given observation results in an outcome of type A or B is constant, independent of the number of observations. 

The occurrence of a type A event in any given observation does not affect the probability that the event A or B will occur again in subsequent observations. 

Examples of such experiments are tossing a coin (heads or tails is the outcome), inspecting a number of similar items for defects (items are defective or not), and picking up objects from a box containing two types of objects. 

The binomial distribution will be introduced with the help of the following experiment. 

In principle, the statistics of radioactive decay are binomial in nature. If we were to toss a handful of coins onto a table and then examine the arrangement, we would find coins in one of two dispositions - heads up or tails up. Similarly, if we could prepare a radioactive source and, during a particular period of time, monitor each individual 

Let us suppose that we could determine exactly which of the atoms, and how many, decayed during the count period. If we were able to repeat the experiment, we would find that different atoms and a different number of atoms decayed in the same period of time. We can regard each such measurement, each count, as a sample in the statistical sense, an attempt to estimate the true decay rate. We would expect the distribution of these counts to fit a binomial distribution (sometimes called a Bernoulli distribution). This distribution applies because  

If we consider each atom in our source there is a certain probability, p, that the atom will decay during the period we choose to make our measurement. This probability is related to the decay constant of the atom and it is straightforward to demonstrate that  

Regardless of the shape of the distribution, the most likely number of decays is given by Equation (5.19)  

Taking the square root of the variance, we can calculate the standard deviation and, for the three specific cases, this would be 1.34, 2.24 and 1.34 decays (or counts, assuming 100% efficiency). Equation (5.19) is interesting in that it predicts that as the probability becomes very small or very near to 1, the width of the distribution or, we might say, the uncertainty on the number of decays, tends to zero. This is not unreasonable. If / = 1, we can expect all atoms to decay and if = 0 none to decay. In either case there is no uncertainty about the number of decays which would be observed. 

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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. 

Binomial Distribution The binomial distribution describes a population in which the values are the number of times a particular outcome occurs during a fixed number of trials. Mathematically, the binomial distribution is given as... 

The binomial distribution describes a population whose members have only certain, discrete values. A good example of a population obeying the binomial distribution is the sampling of homogeneous materials. As shown in Example 4.10, the binomial distribution can be used to calculate the probability of finding a particular isotope in a molecule. 

A portion of the binomial distribution for atoms of in cholesterol is shown in Figure 4.5. Note in particular that there is little probability of finding more than two atoms of in any molecule of cholesterol. 

Portion of the binomial distribution for the number of naturally occurring atoms in a molecule of cholesterol. 

As a starting point, let s assume that our target population consists of two types of particles. Particles of type A contain analyte at a fixed concentration, and type B particles contain no analyte. If the two types of particles are randomly distributed, then a sample drawn from the population will follow the binomial distribution. If we collect a sample containing n particles, the expected number of particles containing analyte, ti, is... 

Figure A6.1 and Table A6.1 show how a solute s distribution changes during the first four steps of a countercurrent extraction. Now we consider how these results can be generalized to give the distribution of a solute in any tube, at any step during the extraction. You may recognize the pattern of entries in Table A6.1 as following the binomial distribution...

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 binomial distribution function is one of the most fundamental equations in statistics and finds several applications in this volume. To be sure that we appreciate its significance, we make the following observations about the plausibility of Eq. (1.21) ... 

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. 

Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probabinty of obseivdng a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials X = number of successes in n trials p = probability of obseivdng a success on any one trial p = x/n, the proportion of successes in n triails Probability Law... 

Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probabihty of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. 

When the value of p is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with X = np when n > 50 while npi < 5. 

The Poisson distribution follows naturally from the discrete binomial distribution already introduced in the craps and the M-out-of-N problem. As N becomes large, the Poisson distribution approximates the binomial distribution... 

The derivation will not be provided. Suffice it to say that the failures in a time interval may be modeled using the binomial distribution. As these intervals are reduced in size, this goes over to the Poisson distribution and the MTTF is chi-square distributed according to equation 2.9-31, where = 2 A N T and the degrees of freedom,/= 2(M+i). 

A lrLL[uently encountered problem requires estimating a failure probability based on the number of failures, M, in N tests. These updates are assumed to be binomially distributed (equation 2.4-10) as p r N). Conjugate to the binomial distribution is the beta prior (equation 2.6-20), where / IS the probability of failure. 

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. 

The binomial distribution can be used to calculate tlie reliability of a redundant syslein. A redundant system consisting of n identical components is a system tliat fails only if more tlian r components fail. Familiar examples include single-usage equipment such as missile engines, sliort-life batteries, and flash bulbs, which are required to operate for one time period and are not reused. Associate "success with tlie failure of a component. Assume tliat tlie n components are independent witli respect to failure, and tliat tlie reliability of each is 1 - p. Then X, tlie number of failures, has tlie binomial pdf of Eq. (20.5.2) and the reliability of the redundtuit system is... 

The binomial distribution applies to random variables where there are only two possible outcomes (A or B) for each trial and where the outcome probability is constant over all n trials. If the probability of A occurring on any one trial is denoted as p and the number of occurrences of A is denoted as x, then the binomial coefficient is given by... 

The median, like the mean, is a measure of the midpoint or center of the distribution. In some cases, such as the gaussian or uniform distribution, the median coincides with the mean, but consideration of the binomial distribution with p i shows that this is not generally true. The... 

In the discussion of hypoelectronic metals in ref. 4, the number of ways of distributing Nv/2 bonds among NL/2 positions in a crystal containing N atoms with valence v and ligancy L was evaluated. The number per atom is the Nth root of this quantity. Structures for which the number of bonds on any atom is other than v-l,v, orv + l were then eliminated with use of the binomial distribution function [only the charge states M+, M°, and M are allowed by the electroneutrality principle (5)]. In this way the following expression for rhypo, the number of resonance structures per atom for a hypoelectronic metal, was obtained ... 

It is evident that the calculation of the number of resonating structures must be made in a different way from that for hypoelectronic metals, because M+ and M form the same number of bonds and are therefore classed together in the calculation of the number of ways of distributing the bonds. We consider first the valence v of a hyperelectronic metal whose neutral atoms form z bonds and whose ions M+ and M" form z + 1 bonds. For any atom, with average valence v, the number of structures, b, having n bonds, is, by the assumption used previously (4), proportional to the probability given by the binomial distribution ... 

The partition of the ten tritons in the fission plane is, of course, different for different fission channels. The curve in Fig. 13, corresponding to random distribution of the ten tritons, has been calculated with the de Moivre approximation to the binomial distribution function. The approximation to the experimental points (21) suggests that good agreement could be obtained by a more refined calculation involving consideration of the various channels for the fission reaction mentioned above. 

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. 

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. 

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 ... 

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

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

The Poisson distribution is actually a special case of the binomial distribution, a fact that is only of mild peripheral interest here, as we will not be using that fact. The formula for the Poisson distribution is... 

The statistics of this process is identical to those pertaining to the tossing of a coin. The mathematics was first worked out with respect to games of chance by de Moivre, in 1733. It is formally described by the binomial distribution. 

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