Random frequency distribution

A frequency distribution cui ve can be used to plot a cumulative-frequency cui ve. This is the cui ve of most importance in business decisions and can be plotted from a normal frequency distribution cui ve (see Sec. 3). The cumulative cui ve represents the probability of a random value z having a value of, say, Z or less. 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

Experiments have been carried out on the mass transfer of acetone between air and a laminar water jet. Assuming that desorption produces random surface renewal with a constant fractional rate of surface renewal, v, but an upper limit on surface age equal to the life of the jet, r, show that the surface age frequency distribution function, 4>(t), for this case is given by ... 

Beads-on-a-string Frequency distributions of number of nucleo somes/template vary with average nucleosome loading between 4 and 8 nucleosomes/template the distributions are broader than random and contain peaks/shoulders, indicating a tendency for correlated nucleosome loading along templates ... 

The mid-range non-random features of frequency distributions (see Ref [37], above) are similar for short-repeat length arrays (172-12) reconstituted with control histones and long repeat-length arrays (208-12) containing hyperacetylated histones ... 

A statistical term referring to a monoparametric distribution used to obtain confidence intervals for the variance of a normally distributed random variable. The so-called chi-square (x ) test is a protocol for comparing the goodness of fit of observed and theoretical frequency distributions. 

The main objective of this experiment was to demonstrate that a peptide lead compound could be used in rational design of a non-peptide library. One of the natural opiates, met-enkephalin, is used as a hypothetical lead compound. The averaged frequency distribution based on four SA runs is obtained (data not shown). Based on this result, 03 had the highest frequency, and the frequencies of A4, Dll, D13, D14, D16, D2, D3, D5, and D9 are also above random expectation. Apparently, 03 appeared in all the reported active peptoids with opioid activity (cf. Table 1). Comparison of the structure of met-enkephalin (Fig. 5) with 03 indicated that 03 is similar to the side chain of tyrosine, which is the N-terminal residue of met-enkephalin. Among other building blocks found more frequently than random expectation, A4, D3, and D13 are present in the reported opioid peptoids (cf. Table 1). Thus, the SA sampling correctly identified four... 

This experiment represents a scenario when an organic lead compound is available. We chose morphine, a known opiate receptor ligand of non-peptide chemical nature, as a hypothetical lead compound. The averaged frequency distribution based on all four SA runs is obtained (data not shown). The most frequent building block was Dll. Building blocks DIO, D12, D14, and 03 were less frequent, but all above random expectation. 

In the Monte Carlo analysis samples are drawn at random from the residue distribution and then from the apple consumption distribution to provide the data points for the intake distribution. This sequence is repeated several thousand times until a smooth intake distribution curve is produced. The intake distribution shown in Fig. 2.5 represents 20,000 samples drawn from the pesticide residues and apple consumption distributions shown in Figs 2.3 and 2.4. The bars represent the relative frequency of each intake level and the line is the cumulative frequency distribution. The distribution is very skewed and it can be seen that the cumulative frequency is nearing 100% when only the mid-point of the distribution is being approached. This means that very high intakes are relatively rare occurrences. 

The outcome of the exposure equation is a dose. This dose varies because of the variability of the components in the equation. The probability distribution of the dose is generally quite difficult to calculate analytically, but can be fairly readily approximated using a Monte Carlo simulation. The simulation consists of numerous iterations. In an iteration, a single value for each component in the exposure equation is randomly sampled from its corresponding distribution. These component values are then substituted into the exposure equation, and the outcome (exposure) is explicitly calculated. The frequency distribution of the calculated values from numerous iterations is the simulated exposure distribution. The exposure equations and the probability distributions of the components are treated as known in the distributional results presented in this chapter. Thus, the simulated exposure distributions reflect exposure variability - but not uncertainty about these equations, the distributions of the components, and related assumptions. This uncertainty and its quantitative impact on the simulated exposure distribution are presented in Sielken et al. (1996). 

Monte Carlo is a probabilistic technique for simulating the outcome of an equation or model involving random variables. The frequency distribution of simulated outcomes is an estimate of the distribution of random outcomes from the equation or model that is being simulated. 

In the terminology of statistics analytically measured quantities (properties, features, variables) are random variables x. Such a variable may, e.g., be density, absorbance, concentration, or toxicity. Hence, repeated measurements (observations) using the same sample, or measurements of comparable samples, do not result in identical values, x but are single realizations of the random variable x. Using the frequency distributions... 

Suppose that we had a very large number of sine waves, all arranged so that they were in phase at time t = 0, with a random (Gaussian) distribution of frequencies. Eventually the waves will start to destructively interfere (cancel), and the amount of time the waves remain in phase depends on the width of the frequency distribution. For a distribution of frequencies with an uncertainty in frequency of 5% (Av = 0.05v0, where vq is the average frequency), it can be shown that the sum looks like Figure 5.13. 

Cherkasov, 2005 a (79) for descriptors only Artificial neural networks (ANN)3 44 (77) Random peptides chosen according to two amino acid frequency distributions Sets A and B contained 933 and 500 peptides, respectively (see text for details, unpublished data) Training and validation within one set, independent testing on second set 1433 Set A models predicted activity with up to 83% accuracy on Set B Set B models predicted up to 43% accuracy on Set A (see text for details) nd... 

Probability distribution models can be used to represent frequency distributions of variability or uncertainty distributions. When the data set represents variability for a model parameter, there can be uncertainty in any non-parametric statistic associated with the empirical data. For situations in which the data are a random, representative sample from an unbiased measurement or estimation technique, the uncertainty in a statistic could arise because of random sampling error (and thus be dependent on factors such as the sample size and range of variability within the data) and random measurement or estimation errors. The observed data can be corrected to remove the effect of known random measurement error to produce an error-free data set (Zheng Frey, 2005). 

A process that appears to be random and is not explainable by mechanistic theory. A stochastic process typically refers to variations that can be characterized as a frequency distribution based on observable data but for which there is no useful or practical means to further explain or make predictions based upon the underlying but perhaps unknown cause of the variation. For example, air pollutant concentrations include a stochastic component because of turbulence in the atmosphere that cannot be predicted other than in the form of an average or in the form of a distribution. 

Weber and Moran Method of Calibration—Another method of calibrating sieves consists in measuring random openings in the sieve and obtaining size-frequency distributions as qutlined in Chapter 3. The openings are measured by filar micrometer or by projection methods. Two screens are then identical if the mean size of the openings and the standard deviations are the same. Weber and Moran (1938) use the coefficient of variation... 

Statistical formulas are based on various mathematical distribution functions representing these frequency distributions. The most widely used of all continuous frequency distributions is the normal distribution, the common bellshaped curve. It has been found that the normal curve is the model of experimental errors for repeated measurements of the same thing. Assumption of a normal distribution is frequently and often indiscriminately made in experimental work because it is a convenient distribution on which many statistical procedures are based. However, some experimental situations subject to random error can yield data that are not adequately described by the normal distribution curve. 

In the analysis of the effect on the calculated quantity of random errors in measured quantities it is unfortunate that the only model susceptible to an exact statistical treatment is the linear one (II). Here we have attempted to characterize the frequency distribution of the error in the calculated vapor composition by the standard methods and have not included a co-variance term for each pair of dependent variables (12). Our approach has given a satisfactory result for the methanol-water-sodium chloride system but it has not been tested on other systems and perhaps of more importance, it has not been possible, so far, to confirm the essential correctness of the method by an independent procedure. Work is currently being undertaken on this project. 

In most analytical experiments where replicate measurements are made on the same matrix, it is assumed that the frequency distribution of the random error in the population follows the normal or Gaussian form (these terms are also used interchangeably, though neither is entirely appropriate). In such cases it may be shown readily that if samples of size n are taken from the population, and their means calculated, these means also follow the normal error distribution ( the sampling distribution of the mean ), but with standard deviation sj /n this is referred to as the standard deviation of the mean (sdm), or sometimes standard error of the mean (sem). It is obviously important to ensure that the sdm and the standard deviation s are carefully distinguished when expressing the results of an analysis. 

If we compare Eq. (XV.2.8) with Eq. (XV.2.3), we see that the latter is about twice as large. This is to be expected because the latter measures the frequency of all A-B encounters, while Eq. (XV.2.8) measures only new encounters. Collins and KimbalP have pointed out that in a diffusion-controlled bimolecular reaction between A and B, the initial rate which can be characterized by a random spatial distribution of A and B decays to the lower rate given by Eq. (XV.2.9). The reason for this is that the reaction tends to draw off the A-B pairs in close proximity and leaves a stationary distribution of A-B which approaches that given by the concentration gradient of Eq. (XV.2.6). The relaxation time for such a decay is of the order of " riB/ir AB, which for most molecular systems will be of the order of 10 sec, or the actual time of an encounter. Noyes has shown that there exist certain experimental systems in which these effects can be observed. We shall say more about them later in our discussion of cage effects in liquids. 

Lasentec Labtec 1000 is a laboratory instrument that covers the size range from 0.7 to 250 pm in 28 size channels. The data are generated as scanned counts an empirical frequency distribution created from classification of chords from randomly oriented particles. Software can convert these chords to a spherical equivalent distribution on the assumption that the chords were generated from an assembly of spherical particles this software contains a filter system to reject improbable data that would tend to skew the distribution to a coarser size. A discrimination... 

Similar results have been obtained by measuring the Raman spectra of partially polymerized crystals and by other spectroscopic techniques. Here, the vibrational frequency of the polymer chain can be used as a probe for the lattice strain in the vicinity of the dispersed macromolecules It should be noted that monomer and polymer are not strictly isomorphous. The mismatch between monomer stacking and polymer repeat of 0.2 A per addition step has to be accounted for by the monomer matrix. The raman frequencies shift accordingly to the lattice changes (cf. Fig. 11) in agreement with the random chain distribution. 

The shape of a frequency distribution of a small sample is affected by chance variation and may not be a fair reflection of the underlying population frequency distribution check this by comparing repeated samples from the same population or by increasing the sample size. If the original shape were due to random events, it should not appear consistently in repeated samples and should become less obvious as sample size increases. 

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