Two-parameter distributions

In plotting on WeibuU paper, a downward concave plot implies a non2ero minimum life. Values for S < can be selected by trial and error. When they are subtracted from each /, a relatively straight line is produced. This essentially translates the three-parameter WeibuU distribution back to a two-parameter distribution. 

Nadav Levanon [21, 22] has applied OS-CFAR to a Weibull distributed background signal and described the results analytically. Blake [26] analysed OS-CFAR in non-uniform clutter. Weber and Haykin [24] have extended OS-CFAR to a two parameter distribution with variable skewness. 

Weber, P., Haykin, S. Ordered statistic CFAR processing for two-parameter distribution with variable skewness, IEEE Transactions on AES, AES-21, 6 (11/1985), pp. 819-821, 1985. 

This is a two-parameter distribution (cf Equation 1). For RP the values of these parameters are... 

To estimate the parameters of a distribution by the method of moments, the moments of the distribution are expressed in terms of the parameters. Estimates for the values of the moments are obtained from the data, and the equations for the moments are solved for the parameters. For a two-parameter distribution, values for the first two moments are needed. 

The strength variability of ceramic materials can be evaluated using Weibull stahshc, which is based on the weakest-link theory, where the more severe flaw results in fracture propagation and determine the strength [69]. The Weibull two-parameter distribution is given by [14] ... 

The exponential distribution was presented in the preceding Section. This distribution is a one-parameter distribution (A). Mathematical statistics uses a large number of distributions, which may serve, for example, to describe empirical data or random processes. Below the probability density functions of several two-parameter distributions are listed, some of which also exist in versions with three parameters. Details are found in [C-l-C-5]. 

Two-parameter and three-parameter distributions are commonly used to fit the temperature cycling data. The two-parameter distribution fits a straight line though the data, whereas the three-parameter distribution fits a non-linear curve through the data. In general, the two-parameter distribution is more conservative than the three-parameter distribution. Best practice is to perform regression analysis to determine which distribution best fits the data. 

In relativistic atomic structure calculations, the two most common models are the uniformly charged sphere and the Fermi two-parameter distribution. The radial density for the uniformly charged sphere is given by... 

For a one-parameter distribution, it suffices to know any moment (higher than the zeroth) to completely define the distribution. The geometric and Poisson distribntions are examples of one-parameter distributions. For a two-parameter distribution, such as the Gaussian, two moments... 

Gaussian (Normal) Distribution The Gaussian or normal distribution is described by the following equations. It is a two-parameter distribution with the mean being a set independent of the variance b. The Gaussian... 

The failure data of area array solder joints (e.g., flip chips, ceramic ball grid arrays) are often fitted to a lognormal distribution. The choice of the distribution selected may depend on the confidence level chosen for the fit. A lognormal distribution may satisfactorily represent failure data at a lower confidence level in some cases. There are two variations for WeibuU distributions the two-parameter distribution and the three-parameter distribution. The third parameter, called the location parameter, represents the minimum time-to-failure. Sometimes the failure data exhibit a slight curvature at a lower failure probability deviating from a two-parameter Weibull distribution. A three-parameter Weibull distribution can be utilized to better fit the data. 

Two-parameter distribution functions exhibited poor fitting capability. All but one of them are in Group 3. 

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

The shape of a normal distribution is determined by two parameters, the first of which is the population s central, or true mean value, p, given as... 

D. I. Gibbons and L. C. Vance, M Simulation Study of Estimators for the Parameters and Percentiles in the Two-Parameter Weibull Distribution, General Motors Research Publication No. GMR-3041, General Motors, Detroit, Mich., 1979. 

The larger the value of n, the more uniform is the size distribution. Other types of distribution functions can be found in Reference 1. Distribution functions based on two parameters sometimes do not accurately match the actual distributions. In these cases a high order polynomial fit, using multiple parameters, must be considered to obtain a better representation of the raw data. 

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. 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

An eminently practical, if less physical, approach to inherent flaw-dependent fracture was proposed by Weibull (1939) in which specific characteristics of the flaws were left unspecified. Fractures activate at flaws distributed randomly throughout the body according to a Poisson point process, and the statistical mean number of active flaws n in a unit volume was assumed to increase with tensile stress a through some empirical relations such as a two-parameter power law... 

NRA exploits the body of data accumulated through research in low-energy nuclear physics to determine concentrations and distributions of specific elements or isotopes in a material. Two parameters important in interpreting NRA spectra are reaction Qvalues and cross sections. 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

Normal distribution The normal distribution is the best known symmetric distribution, and two parameters completely describe the distribution. It often describes dimensions of parts made by automatic processes, natural and physical phenomena, and equipment that has increasing failure rates with time. 

Irani and Callis (Ref 14) used two parameters of the distribution of ground monocalcium phosphate (which follows the commonly used log normal distribution law) namely, Mg and Og, the geometric mean diameter and the geometric standard deviation, to evaluate the precision and accuracy of electro-formed sieves vs sedimentation as a reference procedure ... 

Chapter 9 consists of the following in Sect. 9.2 the physical model of two-phase flow with evaporating meniscus is described. The calculation of the parameters distribution along the micro-channel is presented in Sect. 9.3. The stationary flow regimes are considered in Sect. 9.4. The data from the experimental facility and results related to two-phase flow in a heated capillary are described in Sect. 9.5. 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

The two singlet distribution functions are not in themselves sufficient to characterise the order in a smectic A phase because there is, in general, a correlation between the position of a molecule in a smectic layer and its orientation. We need, therefore, the mixed singlet distribution function P(z,cos ) which gives the probability of finding a particle at position z and at an orientation P with respect to the director [18,19]. At the level of description provided by the order parameters it is necessary to introduce the mixed order parameter... 

---