Statistical distributions parameters

Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22,23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by... 

Random input variable Statistical distribution Parameters... 

Greulich, F.E. (2008). Modeling airtanker flight distance between concurrent fires The development and use of statistical distribution parameters. Forest Science 54(1), 47-57. 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

Computerized Aggregate of Reliability Parameters (CARP) A computer code developed by SAIC to aggregate data sets into a single generic set determine uncertainty bounds (5th and 95th percentiles) fit raw data to statistical distributions and print reports documenting determinations made. 

Oxyfluoroniobates, M2Nb05F, containing trivalent metals (where M = Ti, V, Cr) have the same type of structure [264], except for Cr2Nb05F, which has a tri-rutile type structure. This exception is related to the ordered, rather than statistical, distribution of chromium and niobium cations in the oxyfluoride octahedrons, which leads to a corresponding increase in cell parameter c. 

Laisk, A. (1983). Calculation of photosynthetic parameters considering the statistical distribution of stomatal apertures. Journal of Experimental Botany, 34, 1627-35. 

The converged parameter values represent the Least Squares (LS), Weighted LS or Generalized LS estimates depending on the choice of the weighting matrices Q,. Furthermore, if certain assumptions regarding the statistical distribution of the residuals hold, these parameter values could also be the Maximum Likelihood (ML) estimates. 

The uncertainties in the model inputs were elaborated using the statistic distribution functions for the initial parameters and also the Monte Carlo simulation. 

An RTD curve, for instance, can be represented in algebraic form in more than one way and for different purposes. The characteristic bell shape of many RTDs is evident in the real examples of Figure 5.4. Such shapes invite comparison with some well-known statistical distributions and representation of the RTD by their equations. Or a realistic mechanism may be postulated, such as a network of reactor elements and a type of flow pattern, and the parameters of that mechanism evaluated from a measured RTD. 

Several of the standard statistical distributions are described by Hahn Shapiro (Statistical Models in Engineering, 1967) with mention of their applicability. The most useful models are the Gamma (or Erlang) and the Gaussian and some of their minor modifications. As an illustration of something different the Weibull distribution is touched on in problem P5.02.18. These distributions usually are representable by only a few parameters that define the asymmetry, the peak and the shape in the vicinity of the peak. The moments are such parameters. 

The occupancy parameter of N is 75% that is three atoms are statistically distributed in the four positions. The real stoichiometry corresponds to Mo2N0.75 and in the unit cell there are 11 atoms instead of the ideal value of 12. The Mo position has the free parameterz for which, in this particular case the value 0.258 has been determined. The Pearson symbol is tI12 (11). 

Pharmacokinetic characterization through estimation of PK parameters returns greater value if the parameters of disposition and their statistical distribution in larger... 

In the problem of selecting a distribution for a ID model of variation, there are 2 kinds of variables, namely, 1) the data, which we know and 2) distribution parameters, which will be assigned values based on the data. Here we will often follow statistical terminology by using the term estimation (of parameters) instead of fitting. In statistical terminology, the values assigned to distribution parameters are termed estimates the expressions used to compute estimates are estimators. ... 

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