Population finite

Another complication Is a finite population correction which takes the form of... 

Kimura, M., and Otha, T. (1969). Average number of generations until fixation of a mutant gene in a finite population. Genetics 61, 763—771. 

In the case of spray towers it has been shown by Thornton 10 that ur is well represented by Hod — j) where u0 is the velocity of a single droplet relative to the continuous phase, and is termed the droplet characteristic velocity. The term (1 — j) is a correction to m0 which takes into account the way in which the characteristic velocity is modified when there is a finite population of droplets present, as opposed to a single droplet. It must be seen therefore that for very dilute dispersions, that is as j -> 0, w0(l — j) o- On the other hand, as the fractional hold-up increases, the relative velocity of the dispersed phase decreases due to interactions between the droplets. Substituting for ur, equation 13.32 may be written as ... 

A number of other discrete distributions are listed in Table- 1.1, along with the model on which each is based. Apart from the mentioned discrete distribution of random variable hypergeometrical is also used. The hypergeometric distribution is equivalent to the binomial distribution in sampling from infinite populations. For finite populations, the binomial distribution presumes replacement of an item before another is drawn whereas the hypergeometric distribution presumes no replacement. 

Caprio MA, Tabashnik BE. 1992. Gene flow accelerates local adaptation among finite populations simulating the evolution of insecticide resistance. J Econ Entomol 85 611-620. 

For quasispecies, the error threshold is lower for finite populations because fitness information can be lost through fluctuations in the population as well as a high mutation rate (Nowak and Shuster, 1989 Bonnaz and Koch, 1998). Under the limit of a finite population M, the error threshold has to be modified... 

Van Nimwegen and Crutchfield (1999a) have constructed a theory for the optimization of evolutionary searches involving epochal dynamics. They showed that the destabilization of the epochs due to fluctuations in the finite population occurs near the optimal mutation rate and population size. Under these conditions, the epoch time is only constrained by the diffusion of the population to a neutral network boundary. Often the optimal parameters are very close to the region in which destabilization is an important effect. This emphasizes that, to utilize neutral evolution, it is important to tune the evolutionary parameters (such as mutation rate and population size) so that the time spent in an epoch is minimized without destabilizing the search. 

Kimura, M., and Ohta, T., 1974, Probability of gene fixation in an expanding finite population. PNAS, 71 3377-3379. 

The quantity [ N — n) JNl] is known as a finite population correction -factor. When the number Nl of particles in the lot is very large relative to the number n of particles in the sample, [(V — n)/Ni] 1. The Var(FE) will then be approximately equal to the statistical rel. var. (sample mean). 

In conventional chemical kinetics, time changes of concentrations are described deterministically by differential equations. Strictly, this approach applies to infinite populations only. It is justified, nevertheless, for most chemical systems of finite population size since uncertainties are limited according to some /N law, where N is the number of molecules involved. In a typical experiment in chemical kinetics N is in the range of 10 or larger, and hence fluctuations are hardly detectable. Moreover, ordinary chemical reactions involve but a few molecular species, each of which is present in a very large number of copies. The converse situation is the rule in molecular evolution the numbers of different polynucleotide sequences that may be interconverted through replication and mutation exceed by far the number of molecules present in any experiment or even the total number of molecules available on earth or in the entire universe. Hence the applicability of conventional kinetics to problems of evolution is a subtle question that has to be considered carefully wherever a deterministic approach is used. We postpone this discussion and study those aspects for which the description by differential equations can be well justified. 

Other important questions, however, remain unanswerable within the deterministic theory. Among them are phenomena directly related to finite populations such as the dependence of error thresholds on population size or the mean lifetimes of mutants in populations. The latter quantity is of particular importance for highly fluctuating mutant distributions such as... 

At present the most that can be said analytically about the error threshold in a finite population is the necessary condition [50]... 

NIR analysis is particularly suited to the verification of the identity of packaged clinical supplies because of its nondestructive nature, speed, and low cost. Because every clinical study is a unique event consisting of a finite population of dosage forms, models can easily be generated and validated, and the final-blinded products can be tested the same day that the analysis request is made. 

For example, consider a finite population (N-IO10) where the expected frequency of resistance Is 10 10. With this frequency, the probability that an individual in a population of 1010 is resistant is 0.63 (according to the relationship identified in Table II). 

The relationshps among population size, frequency of resistance, and the probability of resistance occurring in a finite population are illustrated in Figure 1. Clearly the actual occurrence of resistance in a population increases as the population increases. Therefore, the occurrence of resistance (and its subsequent selection) is less likely in small populations than in large populations. 

A statistician calls the infinite number of results that could be obtained and that are described by the probability distribution function (see chapter 1) the population. The distribution of the values of those results is characterized by the population mean /a and the population standard deviation a. The goal of many of our data analysis methods is to estimate l and a from only a few repeated measurements called a sample. (In this respect the definition of population differs from the biologist s view of a finite population of organisms.)... 

The factor [(.V — n)/N], which can be expressed as [1 - (n/N)], reduces the magnitude of the variance of the mean by the sampling fraction when compared to an infinite population value. This reduction factor is called ihe finite population correction factor or fpc, and it indicates the improved quality of the information about the population when n is large relative to N. As n grows larger, the variance of the population mean decreases and becomes zero if n = N, since at this point the mean is known exactly. In many situations fpc has a minimal effect and is usually ignored if n/N < 0.05, and then [(A — //)/A ] is set equal to 1. The confidence interval at P = a, for the estimated mean y, is given by... 

The observations will be obtained from the work crews (approximately 600 employees) that are in nine companies in which there s been carrying out industrial maintenance works. Due to that the activities will be performed by work crews in each company, it has been determined the employees population that must be evaluated in the research to ensure a representative sample of the population. The size of the sample for a finite population in each company has been calculated considering a reliability level of 95% with an error of 5%. [7]. The results are shown in the table 1. 

A probability sample is one for which every unit in a finite population has a positive probability of selection, but not necessarily equal to that of other units, like random (or uniform probability) samples [15]. 

Equation [21] was derived for the case when the sample is very small in comparison with the whole population. For the case when the sample is a substantial part of the population, a so-called finite population correction factor, (1—should be introduced. Here, N is the total number of samples contained in the whole population, the second term of the right-hand side of eqn [19] should be multiplied by this factor. Consequently, eqn [21] takes the following form ... 

Evidently, the finite population correction should be taken into consideration only for relatively small, finite population cases. It can be neglected whenever N is sufficiently large, as then (1 — approaches unity. Equations [21] and [22] can be used for estimating the number of samples necessary to confine the sampling error within +e with the confidence level chosen for the t value (95%, for example), provided the estimated variance within the sample unit is Sj. 

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