Statistical distributions Gaussian distribution

One consequence of Eq. (3-185) is the important result that the sum of a family of statistically independent, gaussianly distributed random variables is again gaussianly distributed. To show this, let... 

A Gaussian curve is characterised by peak position, peak height and peak width. Commonly, the half width is used, i.e. the width at half peak height. We accommodate this convention. In statistics, the Gaussian distribution is usually normalised to unit integral, however, this is not useful in the present context. 

Gaussian function A highly useful function named after mathematician Carl Friedrich Gauss. The familiar bell-shaped function is symmetric and has the property that its integral is 1. In statistics, a Gaussian distribution is called a normal distribution and has the familiar parameters mean ((j.) and standard deviation (a) ... 

There is considerable individual variation in nutrient requirements. It is generally assumed that requirements follow a more or less statistically normal (Gaussian) distribution, as shown in the upper curve in Figure 1.1. This means that 95% of the population has a requirement for a given nutrient within the range of 2 SD about the observed mean requirement. Therefore, an intake at the level of the observed (or estimated) mean requirement plus 2 x SD will be more than enough to meet the requirements of 97.5% of the population. This is the level that is generally called the RDI, RDA, RNI, or PRI. 

The alternative to transforming the experimentally determined MWD is to transform an assumed MWD. Although any form can be used for the distribution, the form used should be such that it can be expressed in terms of parameters supplied with the calibration standards. Both log normal and Guassian distributions have been used to approximate the MWD of unimodal polymers (4). Both distributions were considered here, and statistically the Gaussian distribution was found to better approximate the MWDs of the narrow MWD standards used in this work. As a result the following Gaussian distribution was used. 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

The degree of data spread around the mean value may be quantified using the concept of standard deviation. O. If the distribution of data points for a certain parameter has a Gaussian or normal distribution, the probabiUty of normally distributed data that is within Fa of the mean value becomes 0.6826 or 68.26%. There is a 68.26% probabiUty of getting a certain parameter within X F a, where X is the mean value. In other words, the standard deviation, O, represents a distance from the mean value, in both positive and negative directions, so that the number of data points between X — a and X -H <7 is 68.26% of the total data points. Detailed descriptions on the statistical analysis using the Gaussian distribution can be found in standard statistics reference books (11). 

Distribution models are curvefits of empirical RTDs. The Gaussian distribution is a one-parameter function based on the statistical rule with that name. The Erlang and gamma models are based on the concept of the multistage CSTR. RTD curves often can be well fitted by ratios of polynomials of the time. 

Gaussian Distribution The best-known statistical distribution is the normal, or Gaussian, whose equation is... 

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... 

The root-mean-square error (RMS error) is a statistic closely related to MAD for gaussian distributions. It provides a measure of the abso differences between calculated values and experiment as well as distribution of the values with respect to the mean. 

In general, when a pharmacological constant or parameter is measured it should be done so repeatedly to give a measure of confidence in the value obtained (i.e., the likelihood that if the measurement were repeated it would yield the same value). There are various statistical tools available to determine this. An important tool and concept in this regard is the Gaussian distribution. 

Descriptive statistics quantify central tendency and variance of data sets. The probability of occurrence of a value in a given population can be described in terms of the Gaussian distribution. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

Each oil-dispersant combination shows a unique threshold or onset of dispersion [589]. A statistic analysis showed that the principal factors involved are the oil composition, dispersant formulation, sea surface turbulence, and dispersant quantity [588]. The composition of the oil is very important. The effectiveness of the dispersant formulation correlates strongly with the amount of the saturate components in the oil. The other components of the oil (i.e., asphaltenes, resins, or polar substances and aromatic fractions) show a negative correlation with the dispersant effectiveness. The viscosity of the oil is determined by the composition of the oil. Therefore viscosity and composition are responsible for the effectiveness of a dispersant. The dispersant composition is significant and interacts with the oil composition. Sea turbulence strongly affects dispersant effectiveness. The effectiveness rises with increasing turbulence to a maximal value. The effectiveness for commercial dispersants is a Gaussian distribution around a certain salinity value. 

Methods. Perhaps the best way of dealing with this thorny problem (common to not only °Th/U geochronology, but also the more classical methods of isotope geochronology as well) is to abandon the reliance on a strictly Gaussian distribution of residuals, whether arising from analytical error or geologic complexities. Robusf in the statistical sense implies insensitivity to departure of the data from the initial... 

Here xik is an estimated value of a variable at a given point in time. Given that the estimate is calculated based on a model of variability, i.e., PCA, then Qi can reflect error relative to principal components for known data. A given pattern of data, x, can be classified based on a threshold value of Qi determined from analyzing the variability of the known data patterns. In this way, the -statistic will detect changes that violate the model used to estimate x. The 0-statistic threshold for methods based on linear projection such as PCA and PLS for Gaussian distributed data can be determined from the eigenvalues of the components not included in the model (Jack-son, 1992). 

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