Distributions binomial probability distributions

Because this type of polymerization is a completely random process, with all molecules having equal probability of reacting, the distribution of molecular weights corresponds to the most probable, or binomial, distribution, which is related to the extent of polymerization as follows (Flory, 1953d) ... 

Equation (5.33) differs from that for a random copolymer (most probable or binomial distribution) with a pure crystalline phase, by the last term in the argument of the logarithm. The result embodied in Eq. (5.33) is a perturbation on the melting point equation pertinent to a pure crystalline phase. When e is very large the change in free energy that is involved becomes excessive. The B units will then not enter the lattice and Eq. (5.33) becomes... 

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. 

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. 

The probability of finding an atom of in cholesterol follows a binomial distribution, where X is the sought for frequency of occurrence of atoms, N is the number of C atoms in a molecule of cholesterol, and p is the probability of finding an atom of... 

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. 

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. 

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... 

A related problem is to find the probability of M failures or less out of N components. This is found by summing equation 2.4-9 for values less than M as given by equation 2.4-10 which can be used to calculate a one-sided confidence bound over a binomial distribution (Abramowitz and Stegun, p. 960). 

The cumulative binomial distribution is given by equation 2.5-33, where M is the number of f ailures out of items each having a probability of failure p. This can be worked backH tirLh lo find tlic implied value of p for a specified P(M, p,... 

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 probability distribution of X, tlie number of successes in n performances of tlie random experiment, is tlie binomial distribution, witli pdf specified by... 

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

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

Tlie number of defective items in a sample of size n produced by a certain machine has a binomial distribution with parameters n and p, where p is the probability tliat an item produced is defective. For die case of 2 observed defectives in a sample of size 20, obtain die Bayesian estimate of p if die prior distribution of p is specified by the pdf... 

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 negative binomial distribution defines the probability of the k occurrence of an outcome occurring on the x trial as... 

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 probability function that has been displayed is a very special case of the more general case, which is called the binomial probability distribution. 

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 >, the hypergeometric distribution can be defined. One example is the probability 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. 

The unconditional model treats the sum of all tumors as a random variable. Then the exact unconditional null distribution is a multivariate binomial distribution. The distribution depends on the unknown probability. 

Table 12 illustrates the computation procedure in the case of m = 5 the plant may be envisaged, as in Section V.l, to consist of m cell banks, the quantity j denoting the number of banks switched back into operation. In the specific case of X = p (equi -probability of switching into either direction), Eq. (47) reduces to the binomial probability distribution of selecting j elements out of m identical elements with a single-event probability of xh. 

For further information see Reference 18.] The event might be the presence of any particular attribute in a sample, such as the detection of a pesticide. Only two levels of the attribute are possible, present or not present. If many attributes contribute to the result of an observation, the binomial probability distribution approaida.es a limiting curve whose equation is given by y = (1/ /211) exp[-(2 jx) As... 

Although a variety of probability distributions have been considered for contagious systems, the most successful appears to be the negative binomial. Here a distinguishing characteristic is that is greater than ji. 

By extension, the probability that a given number of actives or fewer should occur in a given distribution is described by the cumulative binomial distribution function, calculated as... 

Binomial (or Bernoulli) Distribution. This distribution applies when we are concerned with the number of times an event A occurs in n independent trials of an experiment, subject to two mutually exclusive outcomes A or B. (Note The descriptor independent indicates that the outcome of one trial has no effect on the outcome of any other trial.) In each trial, we assume that outcome A has a probability P(A) = p, such that q, the probability of outcome A not occurring, equals (1 - q). Assuming that the experiment is carried out n times, we can consider the random variable X as the number of times that outcome A takes place. X takes on values 1, 2, S,---, n. Considering the event X = x (meaning that A occurs in X of the n performances of the experiment), all of the outcomes A occur x times, whereas all the outcomes B occur (n - x) times. The probability P(X = x) of the event X = x can be written as ... 

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... 

For the statistical purest, chi square tests may be used to determine significance levels, binomial distribution for confidence limits on the probabilities. 

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