Variance of distribution

This table represents the upper and lower tails of the distribution of the ratios of ranges of samples taken from a normal distribution of variance oz. If the ratio R / R2 of the ranges R and R2, with size and N2 respectively, is calculated, the probability that it... 

Figure 15 Analysis of dispersion in a high-speed bladed mixer (from Left to Right multiple trajectories starting from the same volume element two examples of distributions of variance (Martin et al., 2007).

Assuming a normal distribution of variance, the number of samples required, to assume at the 95% confidence level that the median is known to 4, is given by... 

For a normal distribution of variance, the spread of data about the mean is described by the probability equation... 

Spectra describe the distribution of variances over different frequency (/) or wave numbers k). Let Sk k) be the wave number spectrum such that... 

Just as the spectra describe the distribution of variance over different freqnencies or wave numbers, the cospectra describe the distribution of covariances over different frequencies or wave numbers. Vertical fluxes of momentum, heat, moisture, and pollutants are proportional to the co-variances and are important in atmospheric dynamics and thermodynamics. They also have important applications to agricultural problems and to oceanography. In order to measure these quantities, instruments must be designed with the proper frequency response to give correct estimates. Area estimates of surface fluxes can be obtained by using remote-sensing sensors in the atmospheric sm-face layer on a scale of kilometers. 

Clearly, when the signal due to the absorbance band falls so low that its ratio to the smallest value of noise at any one wavelength is less than the signal-to-noise ratio at the peak, then no wavelength further away from the peak can be chosen. But the value of noise, and the actual wavelength at which that smallest noise is found, is random and determined by the probabilistic characteristics of the x distribution of variances. 

Figure 9.14 Simulated recombination rate R(f) of geminate e...h pairs. Computer printouts show (a) the coincidence of the data obtained at temperatures 77, 100, 120, 240, 375, and 600 K, (b) the virtual independence of the normalized rate on the wave function overlap factor 2ya, and (c) the influence of the initial pair separation in units of the lattice constant a. The computer system was a sample of cubic symmetry in which the energy of the hopping sites was distributed according to a Gaussian distribution of variance 0.1 eV. (From Ries, B. and Bassler, H., J. Mol. Electron., 3, 15, 1987. With permission.)...

Figure 9.16 Dissociation probabiiity of an e...h pair of initiai separation 24 A in a medium of dielectric constant e = 3.55 at = 10 V cm- as a function of reciprocal temperature. Data points are the result of a Monte Carlo simulation for a discrete system in which the hopping sites are distributed according to a Gaussian distribution of variance a= 0.1 eV the dashed line is the result for a homogeneous medium. (From Albrecht, U. and Bassler, H., Chem. Phys. Lett, 235, 389, 1995. With permission.)...

This command gives a table showing the distribution of variance from the frill dataset on each principal component (Fig. 5). PCA generates as many principal components as there are parameters, but importantly, the majority of variance is represented in the first few components see refs. 1, 2 for further discussion on PCA and variance). In this example, the first three principal components (PC1-PC3) retain 75 % of the variance from the 20-dimensional dataset. As such, PCl-PC3 can be used to construct a set of two-dimensional plots that will allow the visualization of the data in a more intuitive manner while still retaining the majority of the information from the frill dataset. We will therefore focus our remaining analysis on PC1-PC3. 

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

The F statistic describes the distribution of the ratios of variances of two sets of samples. It requires three table labels the probability level and the two degrees of freedom. Since the F distribution requires a three-dimensional table which is effectively unknown, the F tables are presented as large sets of two-dimensional tables. The F distribution in Table 2.29 has the different numbers of degrees of freedom for the denominator variance placed along the vertical axis, while in each table the two horizontal axes represent the numerator degrees of freedom and the probability level. Only two probability levels are given in Table 2.29 the upper 5% points (F0 95) and the upper 1% points (Fq 99). More extensive tables of statistics will list additional probability levels, and they should be consulted when needed. 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

Statistical Criteria. Sensitivity analysis does not consider the probabiUty of various levels of uncertainty or the risk involved (28). In order to treat probabiUty, statistical measures are employed to characterize the probabiUty distributions. Because most distributions in profitabiUty analysis are not accurately known, the common assumption is that normal distributions are adequate. The distribution of a quantity then can be characterized by two parameters, the expected value and the variance. These usually have to be estimated from meager data. 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity... 

In order to compare populations based on their respective samples, it is necessaiy to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effecd, the t statistic characterizes the way in which two sample means from two separate populations will tend to vaiy by chance alone when the population means and variances are equal. Consider the following ... 

FIG. 23-10 Residence time distributions of pilot and commercial reactors. <3 = variance of the residence time distribution, n = number of stirred tanks with the same variance, Pe = Peclet number. 

Because the datay are random, the statistics based on y, S(y), are also random. For all possible data y (usually simulated) that can be predicted from H, calculate p(S(ysim) H), the probability distribution of the statistic S on simulated data y ii given the truth of the hypothesis H. If H is the statement that 6 = 0, then y i might be generated by averaging samples of size N (a characteristic of the actual data) with variance G- = G- (yacmai) (yet another characteristic of the data). 

A measure of the variability of the differences is the variance S, which is the second moment of the distribution of these differences ... 

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

It has been established that the variance of a rectangular distribution of sample... 

Figure 8-38. Residence time distributions of some commerciai and fixed bed reactors. The variance, equivaient number of CSTR stages, and Peciet number are given for each reactor. (Source Wales, S. M., Chemicai Process Equipment—Seiection and Design, Butterworths, 1990.)...

To deal with all kinds of normal distributions of different means and variances, the cumulative distribution is further normalized. This introduces a new variable u = x - ix)/(t. This operation changes a N(p, a) distribution to a N(0, 1) distribution. From Eq. (12.3) the following is obtained ... 

The mean field treatment of such a model has been presented by Forgacs et al. [172]. They have considered the particular problem of the effects of surface heterogeneity on the order of wetting transition. Using the replica trick and assuming a Gaussian distribution of 8 Vq with the variance A (A/kT < 1), they found that the prewetting transition critical point is a function of A and... 

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